optimization techniques for task allocation and scheduling
TRANSCRIPT
Optimization Techniques for Task Allocation and Scheduling in
Distributed Multi-Agent Operations
by
Mark F. Tompkins
Submitted to the Department of Electrical Engineering and Computer Science
In partial fulfillment of the requirements for the degree of
Master of Engineering in Computer Science
at the Massachusetts Institute of Technology
June 2003
© Mark Freeman Tompkins, MMIII. All Rights Reserved.
The author hereby grants to MIT permission to reproduce and distribute publicly paper and electronic copies of this thesis document in whole or in part.
Author__________________________________________________________________
Department of Electrical Engineering and Computer Science May 24, 2003
Certified by______________________________________________________________
John J. Turkovich Member of the Technical Staff, Charles Stark Draper Laboratory
Technical Supervisor
Certified by______________________________________________________________ Munther Dahleh
Professor of Electrical Engineering and Computer Science, MIT Thesis Advisor
Accepted by_____________________________________________________________
Arthur C. Smith Chairman, Department Committee on Graduate Studies
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Optimization Techniques for Task Allocation and Scheduling in Distributed Multi-Agent Operations
By Mark F. Tompkins
Submitted to the Department of Electrical Engineering and Computer Science on May 21, 2003, in partial fulfillment of the requirements for the degree of
Master of Engineering in Computer Science
Abstract This thesis examines scenarios where multiple autonomous agents collaborate in order to accomplish a global objective. In the environment that we consider, there is a network of agents, each of which offers different sets of capabilities or services that can be used to perform various tasks. In this environment, an agent formulates a problem, divides it into a precedence-constrained set of sub-problems, and determines the optimal allocation of these sub-problems/tasks to other agents so that they are completed in the shortest amount of time. The resulting schedule is constrained by the execution delay of each service, job release times and precedence relations, as well as communication delays between agents. A Mixed Integer-Linear Programming (MILP) approach is presented in the context of a multi-agent problem-solving framework that enables optimal makespans to be computed for complex classifications of scheduling problems that take many different parameters into account. While the algorithm presented takes exponential time to solve and thus is only feasible to use for limited numbers of agents and jobs, it provides a flexible alternative to existing heuristics that model only limited sets of parameters, or settle for approximations of the optimal solution. Complexity results of the algorithm are studied for various scenarios and inputs, as well as recursive applications of the algorithm for hierarchical decompositions of large problems, and optimization of multiple objective functions using Multiple Objective Linear Programming (MOLP) techniques. Technical Supervisor: John J. Turkovich Title: Technical Staff, Charles Stark Draper Laboratory Thesis Supervisor: Munther Dahleh Title: Professor of Electrical Engineering and Computer Science, MIT
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Acknowledgments
This thesis was prepared at The Charles Stark Draper Laboratory, Inc., under internal
Research and Development.
Publication of this report does not constitute approval by the Draper Laboratory or any
sponsor of the findings or conclusions contained herein. It is published for the exchange
and stimulation of ideas.
I wish to acknowledge Mr. John J. Turkovich for giving me the opportunity to study at
Draper, for introducing me to autonomous multi-agent systems, and for providing
guidance and detailed feedback during my work on this project.
I also wish to acknowledge Prof. Munther A. Dahleh for providing his expertise on the
ideas that motivated this project, and for patiently reading and reviewing my thesis.
I would also like to acknowledge Dash Optimization, Inc. for graciously providing me
the use of their optimization software, without which much of the work in this thesis
would not have been possible.
I would like to thank my friends Raj, John, and Greg for keeping me motivated with their
constant encouragement and positive reinforcement.
Finally, I would like to thank my family for all of their love and support, and Mary for
proofreading my thesis and for always being there to help me keep my chin up.
Permission is hereby granted by the Author to the Massachusetts Institute of Technology
to reproduce any or all of this thesis.
____________________
Mark F. Tompkins May 21, 2003
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Table of Contents
INTRODUCTION...........................................................................................................11
1.1 PROBLEM OVERVIEW .........................................................................................12
1.2 APPROACH .........................................................................................................14
BACKGROUND..............................................................................................................17
2.1 COMPLEXITY THEORY ........................................................................................17
2.2 MULTIPROCESSOR SCHEDULING THEORY ..........................................................20
2.2.1 Classification of Scheduling Problems .....................................................20
2.3 OTHER SOLUTION TECHNIQUES .........................................................................23
2.3.1 List Scheduling ..........................................................................................24
2.3.2 Dynamic Programming .............................................................................26
2.4 MATHEMATICAL PROGRAMMING .......................................................................27
2.4.1 Linear Programming.................................................................................28
2.4.1.1 Example LP Problem ............................................................................28
2.4.1.2 Simplex Method ....................................................................................30
2.4.2 Mixed Integer-Linear Programming.........................................................31
2.4.2.1 Branch and Bound.................................................................................32
MULTI-AGENT PROBLEM SOLVING .....................................................................35
3.1 PROBLEM SOLVING FRAMEWORK.......................................................................35
3.2 PROBLEM FORMULATION AND DECOMPOSITION ................................................39
3.2.1 Functional Decomposition ........................................................................40
3.2.2 Arithmetic Example ...................................................................................41
3.3 SCHEDULE OPTIMIZATION AND TASK ALLOCATION...........................................45
3.4 HIERARCHICAL PLANNING AND SCHEDULING ....................................................48
3.4.1 Communications Networks vs. Command Hierarchies.............................49
3.4.2 Job Precedence Graphs vs. Objective Decomposition Hierarchies .........50
SCHEDULE OPTIMIZATION.....................................................................................55
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4.1 MOTIVATION ......................................................................................................55
4.1.1 Flexibility ..................................................................................................56
4.1.2 Assumptions...............................................................................................57
4.2 IMPLEMENTATION ..............................................................................................58
4.2.1 Problem Variables.....................................................................................58
4.2.2 Constraints ................................................................................................60
4.3 MULTIPLE OBJECTIVE OPTIMIZATION ................................................................62
4.3.1 Alternate Objective Functions...................................................................62
4.3.2 Multiple Objective Linear Programming..................................................65
4.3.2.1 Determining Target Objective Values...................................................66
4.3.2.2 Determining the Goal Programming Objective ....................................67
4.4 REFINING THE MILP FORMULATION ..................................................................69
4.4.1 Eliminating Binary Decision Variables ....................................................69
4.4.1.1 Discrete Time vs. Continuous Time......................................................70
4.4.2 Refining Constraints..................................................................................71
SIMULATION AND RESULTS....................................................................................75
5.1 SCHEDULING PROBLEM COMPLEXITY ................................................................76
5.1.1 Agent Variation .........................................................................................76
5.1.2 Job Variation.............................................................................................78
5.1.3 Job Parallelism .........................................................................................80
5.1.4 Conclusion.................................................................................................82
5.2 SIMULATION.......................................................................................................82
CONCLUSION................................................................................................................89
6.1 ORIGINAL CONTRIBUTIONS ................................................................................90
6.2 FUTURE WORK ...................................................................................................91
APPENDIX A: MILP FORMULATION.....................................................................93
APPENDIX B: XPRESSMP™ OPTIMIZATION SOFTWARE..............................95
B.1 INPUT FILE FORMAT ...........................................................................................95
B.2 MOSEL PROGRAMMING LANGUAGE .................................................................95
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B.3 XPRESS™ OPTIMIZER ........................................................................................99
B.4 GRAPHING FUNCTIONALITY .............................................................................100
APPENDIX C: AGENT/SERVICE DISCOVERY...................................................101
C.1 AGENT DIRECTORIES........................................................................................101
C.2 FINDING THE SHORTEST PATH BETWEEN AGENTS ...........................................102
REFERENCES .....................................................................................................105
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Chapter 1 Introduction Serious research on task allocation and scheduling problems began in the 1960’s, and has
become a popular research topic in the past few decades as multiprocessor systems have
emerged and demanded efficient parallel programming algorithms. Many studies have
been done on the complexities, classifications, and techniques required for solving these
problems. Due to the inherent complexity of scheduling problems, however, most
researchers have limited their studies to very simple classes of problems and
approximation heuristics for solving them. While these types of problems and their
solution methods are useful for establishing precise mathematical worst-case bounds and
theorems, many of them lack the complexity necessary for modeling any sort of real-
world scheduling problem that might occur in manufacturing, business, or military
scenarios.
For an example, consider a scenario where a military commander needs to acquire data
about potential targets before forming a battle plan. This data could include enemy
locations, weather conditions, logistics, and terrain conditions. Suppose that there are
various human and robotic agents stationed throughout the area of operations. Each
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agent has the ability to conduct information processing tasks such as collect different
types of data, aggregate data, analyze patterns in data, and send their results to other
agents. The commander wants to assign these information-processing tasks to the agents
so that the results are returned to him in the shortest amount of time. The optimal
schedule produced is subject to the processing power/speed of each agent, the available
communication bandwidth between the agents, and the order in which the data must be
collected. This scenario can be modeled in a manner similar to a multiprocessor
scheduling problem, where each agent is modeled as a computing entity that has specific
execution and communication delay times for each job it is assigned.
Many existing scheduling techniques cannot accurately model multi-agent scenarios such
as this one because the problem includes many more parameters than the techniques’
limited problem models take into account. Due to the NP -hardness of all but the
simplest classifications of scheduling problems [17], most researchers have chosen to
develop scheduling algorithms that sacrifice optimality for a heuristic solution that runs
in polynomial time. However, even these heuristic algorithms usually make some
assumptions and simplifications for the problem to make it computationally feasible.
This thesis focuses on a more comprehensive definition of the multi-agent scheduling
problem in the context of a general problem-solving framework, and presents a technique
for computing these schedules that runs in exponential time, but produces optimal
solutions.
1.1 Problem Overview
This thesis examines problems where there is an environment in which one or more
agents are given an objective to complete in the shortest amount of time possible. This is
called minimizing the makespan of a multi-agent schedule. The objective can be divided
into sub-tasks, or jobs, which can be assigned to other agents in the environment. Each
agent offers different services that can be used to complete these jobs. The following
terminology is used to describe this particular class of problems:
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• Agent, a � {A}: Any physical or functional entity that is capable of acting in an
environment, and communicating with other agents in that environment. An
agent can provide a set of services to other agents, and can use these services to
help satisfy an objective. An agent can be a person with an assigned role (such as
a soldier), a piece of software, an autonomous robotic vehicle, or an entity that
represents a collection of other agents.
• Objective: A final goal state to be reached. This thesis looks at situations where
an objective can be achieved by completing a finite set of jobs.
• Job, j � {J}: An individual task that must be completed to help satisfy an
objective. Jobs may be related to one another through a job precedence graph,
and can sometimes be sub-divided into smaller sets of jobs.
• Job Precedence Graph: A directed, acyclic graph (DAG) where each node
corresponds to a job, and each arrow indicates that one job must be completed
before the next job begins. A job precedence graph may correspond to a data-
flow graph, where data from the result of one job is needed as input to another job
before it begins.
• Service: A functionality provided by an agent, which may be the ability to
complete a particular job or set of jobs.
• Job Start Time, Sj: The time at which job j begins execution
• Execution Delay, Dj,a: The time it takes for agent a to execute service that
completes job j. Execution delay depends on both the amount of data that needs
to be processed in order to complete a job, as well as the processing power of the
agent performing the service.
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• Communication Delay, Cj,a,b: The time that it takes to communicate the results
of job j from agent a to agent b. Communication delay depends on the amount of
data being transmitted and the available bandwidth between pairs of agents.
• Release Time, Rj: The time that job j can begin once its predecessors have
completed. If job j’s predecessors are completed before the release time, that
job’s execution still may not begin until the release time has passed.
• Makespan, Cmax: The total time needed to complete a set of jobs that has been
scheduled on one or more agents. Given a set of jobs {J} and agents {A}, the
makespan can be defined as )(max ,
,maxaj
jAaJj
DSC +=∈∈
. In the problems this thesis
explores, finding an optimal schedule means assigning jobs to agents such that the
schedule’s makespan is minimized.
1.2 Approach
The goal of this thesis is to develop a methodology for solving complex scheduling
problems that take many parameters into account. A new approach is presented for
solving the classic multiprocessor scheduling problem using Mixed Integer-Linear
Programming (MILP) techniques. These techniques can be used to compute optimal
solutions to the most complex classes of scheduling problems by taking into account
parameters such as communication delay, job release times, and multiple job precedence
graphs. Most existing algorithms and heuristics only take a few of these parameters into
consideration, and often place limits on the complexity or structure of the problem such
as setting all execution times the same or requiring that the job precedence graph be in
the form of a rooted tree. These heuristics also sacrifice completely optimal solutions in
exchange for approximations in order to achieve a faster running time.
Chapter 2 of this thesis begins by reviewing relevant background information, including
complexity theory, scheduling notation, problem classification, existing algorithms and
heuristics, and Mathematical Programming theory. Chapter 3 presents the general
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framework that is used in this thesis to model problem-solving in a multi-agent system,
and uses theoretical examples to illustrate the formulation, decomposition, and allocation
of tasks to multiple agents. Chapter 4 introduces the MILP technique that is used by this
thesis to solve various multi-agent scheduling problems, and demonstrates how this
algorithm can be used to minimize the makespan of these schedules as well as optimizing
multiple additional objective functions. Chapter 5 examines the scheduling algorithm’ s
run-time performance based on various inputs, and illustrates an example where schedule
optimization is used to choose an optimal agent organization. Chapter 6 discusses
conclusions drawn from this project and possible future research in this domain.
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Chapter 2 Background To understand scheduling problems and techniques used to solve them, it is essential to
understand the theoretical background that these problems are based on. This chapter
provides an overview of numerical complexity theory, scheduling theory and
classification, as well as Mathematical Programming theory, all of which are used as a
research basis for this thesis.
2.1 Complexity Theory
Computational complexity theory deals with the resources required during computation
to solve a given problem. In terms of scheduling problems, we are concerned with the
time required to solve the problems to optimality as they get larger. To examine this, we
must look at the running time of the algorithms developed to solve these problems. The
running time describes the number of operations that must be performed as a function of
the number of input variables to the algorithm. For example, given n points in a plane,
the number of operations required to find the two closest points in the plane is n*(n-1)/2,
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which corresponds to a running time of O(n2), since only the most significant term in the
polynomial needs to be considered.
In general, we want to determine whether an algorithm can be run in polynomial time or
exponential time. A polynomial-time algorithm is one where the running time is
proportional to O(nk) for some constant k, while an exponential-time algorithm has
running time proportional to O(kn). As the number of inputs to a problem increases,
polynomial-time algorithms tend to run much faster than exponential-time algorithms
because they have a slower growth rate. This is not always the case, as a polynomial-
time algorithm that runs in 10e100*n steps takes much longer to solve than an exponential-
time algorithm that runs in 10e-100*2n steps even for relatively large inputs. In general,
however, this distinction holds true.
Different classifications of problem complexity were developed by Cook [5] and Karp
[12] in the 1970’ s to help determine which types of problems were likely to have fast
solution techniques. They define a decision problem as belonging to the class P
(Polynomial) if the algorithm to solve it can be run in polynomial time on a deterministic
machine. A decision problem is considered NP (Non-deterministic Polynomial) if its
solution can be found in polynomial time on a non-deterministic machine, or
equivalently, if a positive or “yes”-solution to the problem can be checked in polynomial
time for correctness. It follows that P ∈ NP , since the solution to any problem in P can
be checked for correctness by simply running the polynomial time algorithm that
generated the solution. Whether or not P = NP, however, is still an open question that
has not yet been proven or disproven [11].
The hardest class of problems to solve in NP are known as NP-complete problems. Cook
[5] defines a problem as NP-complete if it is in NP, and every other problem in NP is
reducible to it in polynomial time. NP-complete problems are considered to be harder
than other problems in NP because, presently, all known NP-complete problems have
exponential running times with respect to the problem size, and are therefore likely not to
be in P.
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The classes P , NP, and NP-complete deal with decision problems, or problems where the
answer is always “yes” or “no”. At first, it may seem that the scheduling problems
presented in this thesis would not fall into the category of decision problems because they
are combinatorial; the answer to a makespan minimization problem is a time value along
with a mapping of jobs to agents. However, it has been shown that almost any type of
optimization problem that can produce a Turing Machine output, including combinatorial
scheduling problems, can be converted into a decision problem [8]. For example,
consider a general problem that outputs n bits of information given an input (such as a
scheduling problem where each bit corresponds to whether a job is assigned to a
particular agent). This combinatorial problem can be converted into a set of decision
problems where each decision problem corresponds to a single bit position in the answer,
indicating whether that bit is 1 or 0. Therefore, solving a decision version of a problem
cannot be more than n times harder than solving the combinatorial version of the
problem.
An NP-hard problem is a combinatorial problem whose decision version is NP-complete.
It has been proven that most classes of scheduling problems, even greatly simplified
ones, belong to the NP-hard class [12,17]. Because there are currently no polynomial-
time algorithms known to solve most of these scheduling problems to optimality, many
researchers have instead focused on developing heuristics or approximation techniques
that lead to sub-optimal answers but run in polynomial time. However, most of these
heuristics only look at simpler versions and special cases of the multi-agent scheduling
problem, and don’ t include a number of various parameters needed to accurately model
many classes of real-world problems. Some examples of these heuristics and their
limitations are given in Section 2.3.
The algorithm presented in this thesis uses Mixed Integer-Linear Programming (MILP)
techniques to solve the most complicated classifications of the multiprocessor scheduling
problem. Using MILP techniques has several advantages. First of all, MILP produces
exact optimal solutions instead of approximate ones. Second, being a general-purpose
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optimization method, software tools such as Dash Optimization’ s XpressMP™ are
available to efficiently solve an MILP problem once it has been formulated (see
Appendix B). The disadvantage of using MILP techniques is that they are NP –hard, and
therefore may be infeasible to use for solving larger scheduling problems. Chapter 5
explores the limits of the problem input size that the MILP scheduling algorithm can
solve in a reasonable amount of time.
2.2 Multiprocessor Scheduling Theory
Much of the work done for this thesis is based on multiprocessor scheduling theory,
which is concerned with optimally allocating sets of agents (or machines) to complete a
set of tasks over time. Many variations of scheduling problems exist, as do techniques
for solving them. Some of these techniques may consider cases where agents are a scarce
resource, while others assume that there are an infinite number of available agents. Other
techniques may assume that execution time for each job on each agent is constant [7], or
that communication time between agents is zero [4]. Many of these algorithms are
heuristics tailored to specific instances of scheduling problems, but it has been shown that
no single one of these approximation techniques works best for every problem instance
[16]. While many of these techniques refer to the computing entities that the jobs are
assigned to as “ machines” or “ processors” , this thesis refers to them as “ agents” , in order
to reflect the fact that scheduling problems are applicable to a wide variety of fields, and
that the “ agents” may represent people or organizations as well as machines.
2.2.1 Classification of Scheduling Problems
The most general case of multiprocessor scheduling problems studied in this thesis
involves a set {J} of n jobs j1…jn, and a set {A} of m agents a1…am, each of which can
perform a subset of the jobs. E(ji,ai) represents the execution time for a job ji on agent ai,
and C(ji,ai,aj) represents the communication delay for sending the results of job ji from
agent ai to agent aj.
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Lawler, Lenstra, and Kan [15] developed a classification technique for modeling
scheduling problems using notation of the form � _� � _� , three variables that specify
different aspects inherent in scheduling problems.
� represents the agent environment, which can consist of a single agent or a set of
PXOWLSOH�DJHQWV�DFWLQJ�LQ�SDUDOOHO��� ∈ {3�4�5`���:KHQ� � �^3�4�5`m, this means that the
problem instance is dealing with exactly m agents.
� � �3�� �Identical parallel agents. Each agent runs at the same speed, therefore
job processing time is independent of whichever agent is chosen. E(ji,ai) = f(ji).
� ��4���Uniform parallel agents. Each agent may run at a distinct speed, so the
job processing time is uniformly related to the speed of each agent. E(ji,ai) = f(ai).
� � � 5�� � Unrelated parallel agents. A job’ s processing time is an arbitrary
function of the job chosen and the agent chosen to run it. E(ji,ai) = f(ji,ai).
� represents the job characteristics, such as precedence constraints and execution costs
DQG�FRQVLVWV�RI�WKH�VXEVHW�^ 1�� 2�� 3`��� 1 ∈ {ø,PREC,TREE}, 2 ∈ {ø,E(j)=1},and 3 ∈
{ø,rj}.
1 = ø: Jobs are not precedence constrained, and may be executed in any order.
1 = TREE: Precedence constraints for jobs have a rooted tree structure.
1 = PREC: Precedence constraints may exist among jobs, and the job
precedence tree may form any type of directed, acyclic graph (DAG).
2 = E(j)=1: All jobs have execution cost of 1 time unit.
2 = ø: Jobs may have distinct, arbitrary execution costs
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3 = rj: Jobs may have set release times before which they may not be processed,
regardless of whether their predecessors have been completed or not.
3 = ø: Jobs have no release times and may be scheduled as soon as their
predecessors are completed.
represents optimality criteria, or the type of objective function to be minimized or
PD[LPL]HG�IRU�WKH�SDUWLFXODU�VFKHGXOLQJ�SUREOHP��� �∈ {Cmax� &i}.
� �Cmax: The objective is to minimize the maximum cost (usually completion
time or makespan) of the schedule.
� � &i: The objective is to minimize the sum of a set of parameters across all
jobs, i.e. the sum of the weighted completion times for each job.
McDowell [17] in his thesis suggests an additional notation that represents
FRPPXQLFDWLRQ�GHOD\�EHWZHHQ�DJHQWV�� �∈ {k,P,J,JP}:
� �N���&RPPXQLFDWLRQ�FRVW�LV�D�FRQVWDQW��RU���LI�WKHUH�LV�QR�FRPPXQLFDWLRQ�FRVW��between all pairs of agents for each job. C(ji,pi,pj) = k.
� �3� Communication cost is a function only of the pair of agents chosen, but
independent of the job (the amount of data transmitted is constant for all jobs, but
other factors, such as the available bandwidth between agents may change).
C(ji,pi,pj) = f(pi,pj).
� �-���&RPPXQLFDWLRQ�FRVW�LV�D�IXQFWLRQ�RQO\�RI�WKH�MRE��DQG�LV�LQGHSHQGHQW�RI�WKH�agents between which the results of the job are transmitted. C(ji,pi,pj) = f(ji).
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� �-3���&RPPXQLFDWLRQ�FRVW�LV�GHSHQGHQW�QRW�RQO\�RQ�WKH�DJHQWV�FRPPXQLFDWLQJ�the results of the job, but also on the job itself. C(ji,pi,pj) = f(ji,pi,pj).
The class of scheduling problems addressed in this thesis is defined as:
R|PREC,rj|Cmax|JP: This problem is to schedule a set of generally precedence-
related jobs on an arbitrary number of agents, with the objective being to
minimize the schedule’ s makespan. Each job has a distinct number of processing
units and each agent has a set amount of processing power, which corresponds to
job execution time being a function of both the job chosen and the agent chosen to
execute it. There may also be distinct communication delay between each pair of
agents, and that delay may also be a function of the job whose results are being
communicated between the agents. Jobs may also have distinct release times.
This is the most complex class of scheduling problems defined by Lawler, Lenstra, Kan,
and McDowell [2,6], and it is also the most difficult to solve. This type of problem can
easily be proven to be NP –hard, based on the fact that most simpler classifications of
scheduling problems such as P| |Cmax|0 have been proven to be NP –hard also [15]. While
the MILP scheduling algorithm presented in this thesis runs in exponential time, it is not
only capable of solving the R|PREC,rj|Cmax|JP problem, but it can also solve every other
class of scheduling problems defined above.
2.3 Other Solution Techniques
Algorithms to solve scheduling problems fall into two different categories: static and
dynamic. Static, or off-line approaches assume that all information about the scheduling
problem is known a priori, and doesn’ t change as the schedule is being computed or
carried out. Using this approach, the complete solution to the schedule is computed
before the schedule actually begins. Static scheduling is most useful for cases when the
algorithm computing the schedule is computationally intensive, or if the schedule is
going to be run multiple times with the same agents and jobs. Dynamic, or on-line
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scheduling, on the other hand, is more flexible because its computations take place as the
current schedule is executing, and it can usually take into account unexpected changes in
the agent organization or job hierarchy. Because the objective of this thesis is to provide
exact optimal solutions to scheduling problems, the algorithm presented is static, and
assumes that all information is known a priori or can be computed beforehand.
Both static and dynamic scheduling problems have been proven to be NP –hard and are
therefore computationally intensive for larger problems. As a result, most scheduling
techniques rely on faster but less accurate heuristics and approximations. Below is a
sample and analysis of some of these existing solution techniques.
2.3.1 List Scheduling
List scheduling is a stochastic approximation technique that was first examined in detail
in 1966 by Ron Graham [9], who looked at the following classification of scheduling
problems: P|PREC|Cmax|0. This formulation assumes that each agent is identical, and
therefore execution delay is a function of the job only. Also, there is assumed to be no
communication delay between agents.
In list scheduling, each job is first assigned a priority level, and is placed into a list that is
ordered by this level. Also, a global time is defined that starts at 0. The algorithm then
runs according to the following steps:
1) The job tree is analyzed and any jobs whose predecessors have been completed or
which have no predecessors are designated as executable.
2) As soon as an agent becomes available (when it has completed all of its previous
jobs and is idle), the highest executable job in the priority list is assigned to it.
3) That job is deleted from the priority list.
4) Steps 1-3 are repeated until there are either no more executable jobs or no more
available agents. The global time is then incremented until either a job or an
agent becomes available.
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Graham proved that this heuristic produces a schedule whose makespan is no more than 2
times the length of the makespan of the optimal schedule [9]. It should be noted,
however, that list scheduling does produce optimal solutions for some very simple classes
of problems [17]. Some examples of these simple classes of problems include:
• P|E(j)=1|Cmax|0. When using an arbitrary number of parallel identical agents with
no communication delay or job precedence constraints and an execution delay
fixed at 1 unit for all jobs, simply scheduling each task on an agent as soon as that
agent is free automatically minimizes the makespan.
• P|TREE,E(j)=1|Cmax|0. This is the same formulation as the previous problem,
except that jobs are now precedence-constrained in a rooted tree structure. List
scheduling can be used again, this time giving preference to the jobs furthest from
the root of the tree to produce a schedule with an optimal makespan. Many variations of the List scheduling algorithm exist, and they differ mainly in the way
in which they assign priority levels to jobs. In general, the strategy is to find the most
critical jobs and place them higher on the priority list. For example, the Modified Critical
Path algorithm [22] assigns priorities to jobs according to the how far they are from the
root of the tree, therefore giving preference to jobs that are likely to be on the critical
path. The critical path is defined as the longest single chain of jobs in the precedence
tree, and its length is a lower bound on the optimal schedule that can be produced.
One of the problems with regular list-based scheduling algorithms is that they don’ t
usually take inter-agent communication delay into account, as pointed out by Kohler [13]
and Sih [21]. List scheduling is a greedy algorithm that attempts to assign all executable
tasks to processors before advancing the global time to the next step. With large
communication delay between agents, however, this can often result in sub-optimal
makespans. For example, in a situation where there is large communication delay and
short execution time, a better makespan may be achieved by scheduling all tasks on one
agent rather than trying to distribute the computation among other agents. McDowell
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also proved that adding even constant communication delay to the simplest types of
scheduling problems (P|E(j)=1|Cmax|K) makes them NP –hard [17].
2.3.2 Dynamic Programming
Dynamic programming is a technique that models a problem as a set of sequential
decisions and a set of states that are reached as a result of these decisions. The objective
is to define an optimal policy, based on Bellman’ s Principle of Optimality [2]:
An optimal policy has the property that, whatever the initial state and the
initial decisions are, the remaining decisions must constitute an optimal
policy with regard to the state resulting from the first decision.
In other words, an optimal policy is a set of rules that are guaranteed to produce an
optimal solution regardless of the starting state.
Engels, Feldman, Karger, and Ruhl [7] developed an optimal scheduling algorithm using
dynamic programming for Pm|TREE,E(j)=1|Cmax|JP problems which is polynomial with
respect to the number of jobs, under the constraint that the communication delay is
bounded by a constant D. Using their algorithm each state is defined by the set {A,P}
where A is the set of executable jobs, and P is a “ window” showing the last D steps of the
schedule. At each state, jobs are considered to be “ active” , “ waiting” , “ scheduled” , or
“ set aside” , and the algorithm essentially finds the shortest path between the states. There
are O(n3mD+2m-1) states, which means that the algorithm is polynomial with respect to the
number of jobs n, but it grows exponentially with D and the number of processors m.
A variation of the List scheduling algorithm, called Dynamic Critical Path scheduling
[14], computes a new critical path at every step of the list-scheduling algorithm and
rearranges the priority levels of jobs. It then selects a suitable agent for each executable
job based on the predicted start time of the “ critical child” of that job.
27
Dynamic programming is a useful technique for conceptualizing scheduling problems
and setting them up for computation, but there is no single method used to solve all
instances of them. Dynamic programming may be used to develop polynomial-time
approximation algorithms or optimal solutions for very simple scheduling classifications,
but to solve schedules to optimality it still runs in exponential time which is no better
than MILP techniques.
2.4 Mathematical Programming
The algorithm presented in this thesis uses Mathematical Programming (MP) techniques
to solve a variety of classifications of scheduling problems. Mathematical Programming
is a very useful tool for solving complex problems such as these that can be modeled as
an objective function with a set of mathematical constraints. A wide variety of research
disciplines currently use MP techniques to aid in complicated decision-making, from
management science to engineering to the military. Because Mathematical Programming
is concerned with finding optimal ways of using limited resources to achieve an
objective, it is often simply referred to as optimization.
In Mathematical Programming, the objective function to be optimized is of the form
MIN (or MAX): f(X1, X2, … , Xn)
where {X1, X2, … , Xn} is the set of decision variables that characterizes the problem.
These variables can represent anything from the number of certain types of products that
need to be produced by a factory to starting times for various jobs in a schedule. The
objective is usually to minimize or maximize a linear function of these variables, subject
to mathematical constraints of the form:
28
f1(X1, X2, … , Xn) ��E1
… .
fk(X1, X2, … , Xn) ��Ek
… .
fm(X1, X2, … , Xn) = bm
{b1, b2, … , bm} is a set of constants and {f1, f2, … , fm} is a set of functions with
parameters {X1, X2, … , Xn} that represent either an equality or inequality constraint.
These functions may be linear or non-linear, and these two types of problems have
different optimization techniques for solving them. This thesis focuses on linear
programming techniques only, but an overview of non-linear programming can be found
in [3].
2.4.1 Linear Programming
Linear Programming (LP) problems are MP formulations where the objective and each
constraint are a linear function of {X1, X2, … , Xn}. Therefore, an LP formulation would
look like this:
MIN (or MAX): c1X1 + c2X2 + … + cnXn
Subject to: a11X1 + a12X2 + … + a1nXn ��E1
… .
ak1X1 + ak2X2 + … + aknXn ��Ek
… .
am1X1 + am2X2 + … + amnXn = bm
2.4.1.1 Example LP Problem
Suppose a company makes two products, P1 and P2, and has two factories that produce
these products. Factory 1 produces 4.5 kg of P1 per day, and 4 kg of P2 per day. Factory
2 produces 7 kg of P1 per day but only 3 kg of P2 per day. Product P1 can be sold for
$2/kg, and product P2 can be sold for $3/kg. Assuming that both factories operate at a
29
constant rate, this means that operating at full capacity the total processing time necessary
for Factory 1 to produce X units of P1 and Y units of P2 would be 29 X +
14 Y. Likewise,
the total time needed to produce X units of P1 and Y units of P2 for Factory 2 would be
17 X +
13 Y. Because it is not possible for the factories to produce negative quantities of
each product, we specify that X ����DQG�<������ �6LQFH�ZH�ZDQW to show that the daily
processing requirements do not exceed production capacity at each factory, the problem
can be modeled as:
Minimize: -2X – 3Y
Subject to: 29 X +
14 Y ���
17 X +
13 Y ���
X, Y ���
The linear constraints and optimal objective function value are modeled in Figure 2-1,
below:
Figure 2-1: A Linear Programming problem and solution
Y
X
Feasible Region
Objective Function
Linear Constraints
Optimal Value X=2.17,Y=2.077
30
In this figure, the shaded region represents the feasible region, or the set of points which,
when chosen, satisfy all of the constraints specified in the problem. As seen in the graph,
the feasible region is a polygon whose edges correspond to the linear constraints. The
objective function line is moved outward (increasing its X and Y intercept values) until it
is tangent to the feasible region at one of its corner points. This point represents the
optimal value for X and Y, the point at which the value of the objective function is
maximized.
Certain LP problem formulations may have multiple or infinitely many optimal solutions,
such as when the objective function is parallel to one of the linear constraints. An LP
problem may also have an unbounded optimal solution, or it may be considered infeasible
when the constraints do not allow for any solution that satisfies all of them.
2.4.1.2 Simplex Method
The Simplex Method is a widely used computational tool for solving LP problems. It is a
fairly simple algorithm, and it has been proven to find optimal solutions (if there are any)
to all instances of LP problems. In addition, as a byproduct it provides useful data, such
as how an optimal solution varies with respect to the problem data and reports when a
solution is unbounded or infeasible. The Simplex Method works because a single
optimal solution to any LP problem always has to occur on a corner point of the feasible
region [3]. Therefore, only these points need to be considered in the search for an
optimal solution.
The Simplex Method works by first converting all constraints that are inequalities into
equalities. It does this by defining an extra positive-valued slack variable for each of
these constraints, subtracting it from each ��FRQVWUDLQW�DQG�DGGLQJ�LW�WR�HDFK���FRQVWUDLQW���Using the previous example, the resulting LP formulation with slack variables S1 and S2
would look like this:
31
Minimize: -2X – 3Y
Subject to: 29 X +
14 Y + S1 = 1
17 X +
13 Y + S2 = 1
X, Y, S1, S2 ���
The constraints are now a series of 2 equations with 4 variables. The Simplex Method
then proceeds to set any 2 of these variables to 0 (or their lower bound), and solves for
the other 2 variables. This solution is guaranteed to be on one of the corner points of the
feasible region, and is called a basic feasible solution.
Rather than enumerating all basic feasible solutions and choosing the best one, the
Simplex Method begins instead at a random corner point, and moves to an adjacent
corner point. It does this only if the solution there is better than the previous one until it
can find no better solution. By using this “ greedy” approach, it is possible to find the
optimal solution using fewer iterations than would otherwise be needed.
2.4.2 Mixed Integer-Linear Programming
Many MP problems exist where it is necessary to restrict the decision variables to integer
or binary values. Examples include cases where the decision variable represents a non-
fractional entity such as people or bicycles, or where a decision variable is needed to
model a logical statement (such as whether or not to assign task A to agent B). These
problems are called Mixed Integer-Linear Programming (MILP) problems, and are often
much harder to solve than LP problems. This is because instead of having feasible
solution points at the easily computed corners of the feasible region, they are instead
usually internal and more difficult to locate. For example, constraining X and Y from the
previous LP formulation to have integer values, the feasible solution points are shown in
Figure 2-2:
32
Figure 2-2: A Mixed Integer-Linear Programming problem showing all feasible integer solutions
The first step in solving a MILP problem such as this is to solve the linear relaxation of
that problem. This simply means removing the constraints that any decision variables
have integer values and solving the resulting LP problem using an algorithm such as the
Simplex Method. The result is one of the following outcomes:
a) The LP problem is infeasible, so the MILP problem is also infeasible.
b) The LP is unbounded and is probably not a well-posed problem.
c) The LP has a feasible solution and all integrality constraints are satisfied,
so the MILP has also been solved.
d) The LP has a feasible solution, but not all of the decision variables have
integer values.
No further work is necessary for outcomes a), b), and c), but more work is required to
find the optimal solution for case d). In most cases the Branch and Bound technique is
used in this case to find the integer solution, as explained in the next section.
2.4.2.1 Branch and Bound
The idea behind Branch and Bound is to divide the feasible region into sub-problems
according to integer values and to use LP techniques recursively to tighten upper and
Y
X
Linear Constraints
Feasible Integer Solutions
33
lower bounds around the optimal solution until they meet or until an LP relaxation results
in an integer solution. After the initial LP relaxation of the problem has been solved, the
resulting decision variable values are analyzed. A non-integer-valued variable is chosen,
and then two sub-problem “ branches” are created which are the same as the previous LP
problem, except that the variable chosen is constrained to be greater than or equal to its
upper integer bound (rounding the decimal solution up) in one case, and less than or
equal to its lower integer bound (rounding the decimal solution down) in the other case.
Each of these sub-problems is recursively solved and “ branched” again using LP
techniques and choosing a different variable, until either:
a) The LP problem of the branch is infeasible,
b) The solution to the LP problem is integer-valued, or
c) The solution to the LP problem is worse than a previously computed
solution.
In a maximization problem, the optimal objective function value of the LP relaxation is
always an upper bound on the optimal integer solution. Likewise, any integer feasible
point found is always a lower bound on the optimal integer solution [3]. These values are
used to update the upper and lower bounds at every step, narrowing down the possible
solutions until the optimal integer solution is found.
Branch and Bound works better for MILP problems than completely enumerating every
possible integer solution, but its running time may still grow exponentially with respect to
the problem size in some cases. The total running time of Branch and Bound depends
mostly on how well the LP relaxation approximates the convex hull of the feasible integer
set. The convex hull of a feasible set of points S of a MILP is defined as the smallest
polyhedron that contains S. Mathematically, the convex hull for S is defined as:
C =
=≥∈ ∑∑= i
iiii
K
ii Sxx 1,0,
1
λλλ (2-1)
The closer the LP relaxation is to this bound, the less time it takes for Branch and Bound
to find the optimal integer solution.
34
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35
Chapter 3 Multi-Agent Problem Solving In order to provide a context for the multi-agent scheduling algorithm presented in this
thesis, this chapter presents a general problem-solving framework that explores job
decomposition, optimal task allocation, and solution formulation. In addition, examples
from various problem domains are used to illustrate each of these techniques.
3.1 Problem Solving Framework
The process of multi-agent planning and problem solving can be modeled by a 4-phase
closed-loop command and control architecture [10], shown in Figure 3-1(a). This
architecture contains the following sub-functions: Plan, Execute, Monitor, and Diagnose.
The arrows represent the flow of data between each of these functions, which forms a
continuous feedback mechanism.
36
Figure 3-1(a): Problem solving framework represented by a closed-loop command and control architecture.
Figure 3-1(b): Hierarchical application of the problem-solving framework in (a).
Plan
ExecuteMonitor
Diagnose
Objective
Sub-ObjectivesSub-Solutions
Solution
Plan
ExecuteMonitor
Diagnose
Objective
Sub-ObjectivesSub-Solutions
Solution
Plan
ExecuteMonitor
Diagnose
Plan
ExecuteMonitor
Diagnose Plan
ExecuteMonitor
Diagnose
Solution Objective
Sub-Objective BSub-Objective A
Plan
ExecuteMonitor
Diagnose Plan
ExecuteMonitor
Diagnose
Plan
ExecuteMonitor
Diagnose Plan
ExecuteMonitor
Diagnose
Solution Objective
Sub-Objective BSub-Objective A
37
Figure 3-1(b) shows a hierarchical application of this closed-loop architecture, where the
top level can be seen as a controller that decomposes a problem into two sub-problems,
and sends these sub-problems to be planned, executed, monitored, and diagnosed in a
similar fashion. This hierarchy can be extended to multiple levels depending on how a
problem is decomposed.
Each of the functions in Figure 3-1(a) can be further decomposed to address the more
specific problem of multi-agent planning and scheduling, as shown in Figure 3-2. In this
figure, the specific tasks of job decomposition, job/agent mapping, optimal schedule
computation, task allocation, and solution formulation are represented within this higher-
level architecture.
Figure 3-2: Multi-agent planning and scheduling process in the context of the closed-loop command and control architecture.
Objective
Sub-ObjectivesSub-Solutions
Solution
Decompose into individual jobs
Map jobs toavailable
agents/services
Agent/servicediscovery
Compute optimalschedule
Allocate jobsto other agents
Collect completedjob data from
agents
Formulatesolution
Send results to appropriate agents
Plan
ExecuteMonitor
DiagnoseData Flow
PrecedenceRelations
Objective
Sub-ObjectivesSub-Solutions
Solution
Decompose into individual jobs
Map jobs toavailable
agents/services
Agent/servicediscovery
Compute optimalschedule
Allocate jobsto other agents
Collect completedjob data from
agents
Formulatesolution
Send results to appropriate agents
Plan
ExecuteMonitor
DiagnoseData Flow
PrecedenceRelations
38
As shown in Figure 3-2, these steps consist of:
• Formulating or receiving from another source a global objective to be achieved.
• Structuring this objective in a form where it can be easily decomposed into a
partially-ordered set of sub-problems or jobs.
• Surveying the environment for available agents and services that may be used to
complete these jobs.
• Mapping jobs to available services or sets of services that are capable of
completing them.
• Determining the allocation of jobs to agents, such that the resulting schedule is
optimized according to user-defined parameters.
• Formulating the solution based on results from all participating agents, and
forwarding this solution to the appropriate agents.
It is important to note that the small arrows between the specific tasks in Figure 3-2
represent precedence relations, while the large arrows between the high-level functions
represent continuous data flow and feedback between these phases. It should be noted,
however, that the feedback provided by the data flow could represent precedence
relations at a higher level itself in the context of this architecture, as one phase may not
be allowed to begin its next cycle until it has received data from another phase. In this
case, to represent this high-level data flow in the form of precedence constraints it is
necessary to view the feedback graph in Figure 3-1(a) as a continuous cycle that runs in
multiple iterations. Therefore, in order to show precedence relations between the 4 main
functions each iteration must be shown separately, with the feedback arrows pointing to
the next iteration of each function instead of back to the original version. Figure 3-3
shows an example of these precedence relations between a 2-level deep hierarchy of
command and control nodes; it should be noted that the four functions in each node are
shown in a different orientation to provide a clearer picture of the precedence constraints
between each iteration. These high-level precedence relations in the command and
control architecture are explored further in a scheduling application example in Chapter
5.
39
Figure 3-3: Precedence relations between each main function in the closed-loop hierarchy, shown in multiple iterations to represent feedback between functions at each stage.
The following sections describe the planning process from Figure 3-2 in detail, including
finding the allocation of jobs to agents such that the resulting schedule is optimized,
which is the main focus of this thesis.
3.2 Problem Formulation and Decomposition
In the multi-agent framework that we are considering, there is a set of heterogeneous
agents {A} in a network, each with a different set of available abilities and services. A
single agent is given an objective to complete, possibly from another agent, and it wishes
to take advantage of the resources provided by these other agents in the network to
complete the objective more efficiently. The planning agent’ s first step is to decompose
its objective into a set of jobs {J} that can be allocated to other agents in the network and
completed in parallel.
Diagnose2
Monitor2Execute2
Plan2
Diagnose2
Monitor2Execute2
Plan2
Diagnose1
Monitor1Execute1
Plan1
Objective
Diagnose1
Monitor1Execute1
Plan1
…
Iteration 1 Iteration 2
HigherLevel
LowerLevel
Diagnose2
Monitor2Execute2
Plan2
Diagnose2
Monitor2Execute2
Plan2
Diagnose1
Monitor1Execute1
Plan1
Objective
Diagnose1
Monitor1Execute1
Plan1
…
Iteration 1 Iteration 2
HigherLevel
LowerLevel
40
However, there are often many possible problem formulations for a given objective and
choosing the best way to decompose the objective may depend on the structure of the
agent organization and the number of different service types provided by these agents. In
the scenario that this thesis addresses, the planning agent’ s main goal is to choose the job
precedence graph that produces the schedule with the shortest possible makespan.
3.2.1 Functional Decomposition
Large-scale problems from many different domains can be decomposed by defining them
using generalized functions of the form {F(x), G(y), H(z) … }. A computer program or
systems engineering problem may be described this way where each function
corresponds to an atomic operation or set of operations. These operations may be
ordered relative to each other or may iterate through other operations recursively. As an
example, consider the following control system for an unmanned autonomous vehicle
(UAV) shown in Figure 3-4.
Figure 3-4: Internal control and navigation system of an autonomous vehicle.
Camera
SteeringMechanism
Transmission
Engine
Actuators
GPS
Compass
MissionPlanner
Navigation
Heading
Throttle
Direction
Steering
Pan
Zoom
Guidance
Sensors Control
(x,y)
Target
(XR,YR, R ) T
D
S
Z Camera
SteeringMechanism
Transmission
Engine
Actuators
GPS
Compass
MissionPlanner
Navigation
Heading
Throttle
Direction
Steering
Pan
Zoom
Guidance
Sensors Control
(x,y)
Target
(XR,YR, R ) T
D
S
Z
41
Each node in the Control section of Figure 3-4 represents an internal procedure with a
specific input and output, and the arrows between these procedures represent data
dependencies. These processes can be seen as abstract functions, with the data being
passed between them as the outputs and inputs to each function. Suppose the first letter
of each process is used to name the function that it performs, i.e. Guidance output
(XRYR, R) = G(WDUJHW�[�\� ). In this case multiple data dependencies between processes
can be represented using nested functions, i.e. the Camera Zoom setting would be defined
as Z = Z(H( ),G(target,N( ),H( ))).
These functions can be used to form different precedence-constrained chains of jobs that
are used to achieve the same objective. For example, if the objective given to the UAV is
to observe a certain point (x1,y1) on a map with its camera, it may choose to formulate
and decompose a plan to complete this objective in a variety of ways. One possibility
would be to change the camera’ s pan and zoom settings to point at the new target (x1,y1)
without moving the vehicle itself, in which case the functions used to complete the
objective could be modeled as
Camera: {Z(H( ), G(target,N( ),H( ))),P(H( ))}.
An alternative approach to this may be to move the vehicle in a straight line to a new
position such that the camera is pointed at the target without changing the actual camera
settings. In this case the functional decomposition would instead be:
Transmission: {T(G(target,N( ),H( )),N( )), D(G(target,N( ),H( )),N( ),H( ))}
Engine: {T(G(target,N( ),H( )),N( ))}
These two functional decompositions and others may be used to achieve the same
objective, and choosing which one to use may depend the efficiency and reliability of
each method, as well as a number of external environmental factors such as terrain
conditions and visibility.
3.2.2 Arithmetic Example
Arithmetic problems are another domain where multiple functional decompositions may
be used to achieve a single objective, or in this case a solution to a compound problem.
The operations add (x+y), subtract (x-y), and multiply (x*y) can be modeled by the
42
functions F(x,y), G(x,y), and H(x,y) respectively. Each of these functions obeys different
mathematical laws, which allow the same arithmetic problem to be decomposed multiple
ways. For example, the Commutative Law applies to multiplication and addition, and
states that F(x,y) = F(y,x). Similarly, the Associative Law states that F(F(x,y),z) =
F(x,F(y,z)). Finally, the Distributive Law states that H(x,F(y,z)) = F(H(x,y),H(x,z)).
Suppose that in a particular environment there are multiple agents that can provide three
possible services: add, subtract, or multiply. After one agent formulates and
conceptualizes an arithmetic problem, it may then decompose the problem and send the
sub-problems to other agents to solve in parallel. Determining which agents to send the
problem to may depend on a number of factors such as available bandwidth, physical
proximity, and processing speed of each agent. Figure 3-5 shows this agent network.
Figure 3-5: An agent network, indicating execution and communication delays and available services.
In Figure 3-5, suppose that Agent A is given the responsibility to compute the solution to
the problem (5+8)*3 + (6-4)*(8+2). Agent A does not have the ability to add, subtract, or
Agent A Services/Delay: Initiate (I) / 1s Terminate (T) / 1s
(5+8)*3 + (6-4)*(8+2)
Agent C Services/Delay: Multiply (*) / 2s
Agent G Services/Delay: Add (+) / 1s Multiply (*) / 1s
Agent E Services/Delay: Subtract (-) / 1s
Agent B Services/Delay: Add (+) / 3s
Agent D Services/Delay: Add (+) / 2s
Agent F Services/Delay: Multiply (*) / 1s
4s
2s
1s
2s
2s
5s
2s
3s
43
multiply, so it determines that it must seek out the services of other agents to solve the
problem. The agents are connected to one another according to the communication
network shown in the figure, and each other agent provides one or more of these services.
The weights on each edge in the graph represent the aggregate time needed to send data
over that link. This model assumes that the results of each job contain the same amount
of data, so communication delay is a function of available bandwidth between agents
only, and is independent of the job data being transmitted. The services offered by each
agent are shown, as well as the execution delay required to perform each service. Agent
A’ s services, Initiate and Terminate, reflect the fact that data must be sent from Agent A
to all other agents before the rest of the problem can begin, and that the final solution to
the problem must be sent back to Agent A to be processed.
Using the functional notation defined above, Agent A’ s problem can be written as
F(H(F(5,8),3),H(G(6,4),F(8,2))). Using the Distributive Law, however, the same
problem can be decomposed as G(F(H(5,3),H(8,3),H(6,8),H(6,2)),F(H(4,8),H(4,2))), or
(5*3 + 8*3+6*8 + 6*2) – (4*8 + 4*2). The job precedence graphs for these two
equivalent formulations are shown below in Figure 3-6.
(a) (b)
Figure 3-6: Two possible formulations and job precedence graphs for the same arithmetic problem.
*5*4*3*2*1 *6
+7 +8
Initiate0
Terminate10
(5*3 + 8*3 + 6*8 + 6*2) – (4*8 + 4*2)
Terminate7
Initiate0
+1 +3
+6
*4 *5
(5+8)*3 + (6-4)*(8+2)
-2
-9
*5*4*3*2*1 *6
+7 +8
Initiate0
Terminate10
(5*3 + 8*3 + 6*8 + 6*2) – (4*8 + 4*2)
Terminate7
Initiate0
+1 +3
+6
*4 *5
(5+8)*3 + (6-4)*(8+2)
-2
-9
44
In Figure 3-6, each node represents an individual job that must be completed and the
arrows represent precedence constraints between the jobs. For example, in the Figure
3-6(a), job (+1) must be completed and the data sent to the agent assigned to process job
(*4) before that job can begin execution. Each job is identified by the service needed to
complete it (add, subtract, or multiply), and a unique identification number. Once a job
precedence graph has been constructed, agents in the network must be identified that
have the ability to complete each of these jobs using their provided services, and
communication delays must be computed between each pair of agents (see Appendix C).
In some cases the services used to complete a job may differ from agent to agent, and in
other cases a combination of services may be required for more complicated jobs.
One strategy for determining the best problem decomposition is to look at each different
formulation, calculate the optimal agent allocation for it, and choose the decomposition
that produces the shortest schedule. However, since there are often many different
possible job precedence graphs for any single objective and calculating schedules for
each of them is an NP-hard problem as shown in Chapter 2, it may not be feasible to
calculate each one individually. In this case a heuristic technique may be used based on
the properties of the agent/job ratio and the structure of the agent network. Some of these
possible techniques include:
• Choosing the job precedence graph with the smallest number of nodes
(minimizing the total number of operations to be performed). This would be the
best choice if there were a small agent/job ratio or large communication delays,
and the number of available agents and services was a limiting factor in the
problem. It follows that in an environment with limited computational resources,
the best problem formulation is usually the one with the fewest computations.
• Choosing the job precedence graph with the shortest depth (minimizing the
critical path). This would be the best choice if there was a large agent/job ratio
and communication delays between agents were small. With computational
resources no longer being a limiting factor, the optimal problem formulation
would be the one with the most jobs performed in parallel on different agents.
45
• Choosing the job precedence graph whose jobs most closely correspond to the
number of corresponding service types offered by different agents in the network,
or to the agent structure itself.
While none of these strategies is guaranteed to produce the optimal problem formulation,
they are all likely to rule out many of the obviously sub-optimal job precedence graphs.
It should be noted that there are other approaches to this problem where instead of
formulating the problem to best fit the agent organization, the organization is modeled
around the problem structure. As Rifkin [20] points out,
“…there is an inextricable inter-dependence between structure of the work and structure
of the team. The structure of the work is optimal relative to the structure of the team, and
the structure of the team is optimal relative to the particular structure of the work.”
Because of this interdependency between the agent organization and the problem
structure, there is currently no single best algorithm for determining either; heuristics are
used instead to calculate one based upon the other. Future work on this topic may focus
on developing optimization techniques for finding the best job decomposition for an
agent network, but that is beyond the scope of this thesis.
3.3 Schedule Optimization and Task Allocation
After choosing an acceptable job decomposition and gathering all relevant information
about the agent network, a problem-solving agent must determine which other agents to
allocate tasks to. When computing an optimal schedule, an agent may be trying to
achieve one or more of the following objectives:
• Minimize the makespan of the schedule
• Minimize the weighed completion time of each job
• Minimize the total number of agents used to solve the problem
• Minimize the amount of communication bandwidth used
46
• If there is a value associated with assigning a task to a particular agent, maximize
the aggregate value
• If there is a cost (other than time) associated with assigning a task to a particular
agent, minimize the aggregate cost
The first of these objectives, minimizing the schedule’ s makespan, is the subject of most
existing research on multiprocessor and job-shop scheduling problems. The makespan is
also the primary focus of the algorithm presented in this thesis, but Multiple Objective
Linear Programming (MOLP) approaches are presented in Chapter 4 that can be used to
optimize the scheduling problem with respect to other objectives as well.
Continuing the arithmetic example from the previous section, assume that Agent A
chooses the smaller job precedence tree from Figure 3-6(a): (5+8)*3 + (6-4)*(8+2).
Agent A then wants to compute the schedule with the optimal makespan based on the
agent organization in Figure 3-5. There are many ways that Agent A can solve this
problem. The simplest way would be to have Agent A start at the end of the job
decomposition tree and work backwards, sending the top-level tasks to the closest
available agents that have the ability to complete those tasks. Those agents would then
divide up the next level of tasks and decide which other agents to allocate those tasks to
in a recursive manner, until the bottom of the precedence tree is reached and the solution
is sent back to Agent A. This technique, known as hierarchical planning, is useful in
cases where jobs in the precedence graph are more general near the top and more specific
near the bottom, and is explored in more detail in Section 3.4.
While this method requires the least amount of planning and works well in dynamic agent
environments, it makes the assumption that each agent in the network is capable of
making the same kinds of task allocation decisions that Agent A can. Also, hierarchical
planning may not always result in a schedule with an optimal makespan, because each
agent does not see the entire problem horizon and therefore may not know the best way to
coordinate with other agents.
47
By planning “ routes” for the various jobs to travel in the agent network in advance, a
better schedule can be obtained. For example, Agent A can start at the bottom of the job
precedence graph in Figure 3-6(a) and send those tasks out to the appropriate agents
along with instructions for those agents to send the results of that computation to another
agent, until the final result is sent back to Agent A. MILP techniques can be used to
compute this scheduling problem to optimality and are described in detail in Chapter 4.
Having an agent plan the schedule completely in advance makes it lose the ability to
adapt its plan to dynamic changes in the network during execution of the schedule, but
allows for the makespan of the schedule to be minimized. Figure 3-7 shows the optimal
schedule for the arithmetic example described in this chapter, computed using
XpressMPTM optimization software (see Appendix B). In this Gantt chart, the horizontal
arrows represent job execution times, the diagonal arrows represent communication delay
between agents, and each job is labeled by its unique identifying number defined in
Figure 3-6(a).
Figure 3-7: A Gantt chart showing the optimal schedule for the arithmetic problem described in this
section.
This schedule is very sensitive to changes in both the agent network and the problem tree
structure. For example, if the communication delay between Agents D and G is changed
G
F
E
D
C
B
A
Agent
Time(s)
48
from 5s to 1s (see Figure 3-5), the optimal makespan is reduced from 19s to 15s, and
results in a completely different optimal job allocation, as shown in Figure 3-8:
Figure 3-8: A Gantt chart showing the new optimal schedule computed after changing the communication delay between Agent D and Agent G from 5s to 1s.
The above example demonstrates the value of being able to re-compute the optimal
makespan of a multi-agent schedule when small changes are made to the network or job
structure. If this were a multiprocessor scheduling problem and the same schedule was to
run successively with the same inputs each time, this recalculation would result in a 21%
increase in efficiency.
3.4 Hierarchical Planning and Scheduling
As mentioned in Chapter 2, using MILP techniques for schedule optimization is an NP-
hard problem, and therefore is not feasible for solving large-scale scheduling problems.
Fortunately, under certain circumstances these large problems can be broken down into
smaller sets of scheduling problems that have a shorter problem horizon and can be
solved recursively. This hierarchical planning technique mentioned in the previous
G
F
E
D
C
B
A
Agent
Time(s)
49
section can be useful in situations where there is a large problem that needs to be broken
down in stages, or if agents are organized in a command hierarchy.
3.4.1 Communications Networks vs. Command Hierarchies
In some scenarios, an agent organization may be in the form of a command hierarchy.
For example, the military uses command hierarchies to efficiently group agents with
similar or complementary abilities together, to maintain flexible and robust control over
large numbers of units, and to facilitate information flow throughout the organization.
It is important to distinguish between a command hierarchy and a communications
network. A command hierarchy does not necessarily show all of the communication
paths between agents; the arrows in a command hierarchy instead represent which agents
have the authority to allocate jobs to other agents. In fact, each agent in this hierarchy
may not even be aware of which agents are at different levels in it. For example, in
Figure 3-9 the Unit Commander may not know the names and specific abilities of every
individual soldier and Unmanned Autonomous Vehicle (UAV) the bottom of the
command chain. Agents at each level of the military command hierarchy often only have
the ability to share intelligence with other agents in the same functional group; thus each
level of the command hierarchy can be seen as a unique communication network.
50
Figure 3-9: An example military command hierarchy, adapted from Baxter and Hepplewhite [1]. 3.4.2 Job Precedence Graphs vs. Objective Decomposition Hierarchies
In the diagram in Figure 3-9, a single high-level objective is given to the Unit
Commander, who splits the objective up into lower-level objectives and passes them to
the Sub-Unit Commanders. The Sub-Unit Commanders then each take their objective,
divide it up further, and pass those low-level objectives to their troops, tanks, and UAV’ s.
The Unit Commander’ s objective is usually more general and is considered over a longer
time horizon, while the low-level objectives are more short-term and specific. Higher-
level objectives are said to have large granularity, while lower-level ones have small
granularity.
However, in the job precedence graphs used in the MILP scheduling algorithm presented
in this thesis, such as the arithmetic example presented earlier, jobs typically have very
similar granularities. Each servive (add, subtract, multiply) is functionally independent
Unit Commander
Sub-Unit Commander
Sub-Unit Commander
Sub-Unit Commander
Troop Tank UAV
Higher Level Objective
Lower Level Objectives
Command and Control Hierarchy
Communication Network
51
of the other services, i.e. no service consists of completing a set of any of the others, and
they all have the similar time horizons. How does this hierarchical objective
decomposition fit into our problem-solving model then?
Figure 3-10: Part of an objective decomposition hierarchy for a Military Operations in Urban Terrain (MOUT) scenario.
Figure 3-9 shows an example of an objective decomposition hierarchy in a Military
Operations in Urban Terrain (MOUT) scenario. Suppose that the top objective, “ Survey
Area” , is assigned to the Unit Commander in Figure 3-9. The Unit Commander
decomposes this general objective into a set of smaller, more specific objectives, and
wants to find the optimal Sub-Unit Commanders to assign this next level of objectives to
so that the tasks are completed in the shortest amount of time. In order to compute the
optimal schedule for these jobs, two inputs are needed: an agent network with known
communication and execution delays, and a partially-ordered job precedence graph.
Secure Area
Survey Area
Search for Mines
Infiltrate Building 2
Guard Perimeter
Satellite Data
Search for Snipers
Infiltrate Building 1
Body Heat Sensors
UAV Cameras
Secure Perimeter
Enter Building
Secure Rooftop
52
The jobs at each level in Figure 3-10 have similar granularities and may have additional
precedence constraints relative to each other that aren’ t shown in the figure. For
example, consider the second level of this tree. The four jobs there may have to be
completed in a partial order: The area should be surveyed for dangerous elements before
entering any of the buildings, and only after both buildings are secure should another set
of agents be set to guard the area perimeter. Therefore, these four objectives form their
own job precedence graph:
Figure 3-11: A job precedence graph for the second level of tasks from Figure 3-10.
Because the Sub-Unit Commanders belong to the same network, communication delays
may be computed between each of them. Execution delays may be harder to calculate
though, especially higher up in the decomposition hierarchy, because the jobs are still
very high-level and not very specific. For example, if the Unit Commander asks a Sub-
Unit Commander how long it would take to “ survey the area” , the best that the Sub-Unit
Commander may be able to do is provide an estimate based on the decomposition and
optimal allocation of that sub-objective to other agents. Likewise, the Sub-Unit
Commander can only calculate its own optimal schedule after obtaining execution time
estimates from the agents directly below it. Eventually, these requests for execution
times reach the bottom of the agent hierarchy and the jobs are specific enough that the
agents there can determine their own execution times. The data then flows back up the
hierarchy, enabling schedules to be optimized at each successive level until the Unit
Survey Area
Guard Perimeter
Infiltrate Bldg. 2 Infiltrate
Bldg. 1
53
Commander has enough time estimates to compute its own optimal schedule at the
highest level. Figure 3-12 shows how the problem-solving model described in Section
3.1 may be applied recursively in this manner to optimize task allocations at each level of
an objective decomposition hierarchy.
Figure 3-12: The problem model from Figure 3-2 can be applied recursively to compute optimal schedules each level of an objective decomposition hierarchy.
While this technique is limited to problems where the agent organization is a command
hierarchy and the objective can be decomposed at each level of this hierarchy, it is
nonetheless very useful for breaking up a large scheduling problem into a set of smaller
schedules that are more feasible to solve. Exact limits to number of jobs and agents that
the MILP algorithm presented in this thesis can solve in a reasonable amount of time are
explored in Chapter 5.
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55
Chapter 4 Schedule Optimization
This chapter presents a Mixed Integer-Linear Programming (MILP) formulation that can
be used to optimally solve the types of multi-agent scheduling problems presented in
Chapter 3.
4.1 Motivation
The primary motivation behind developing the MILP formulation in this chapter is to
provide a scheduling algorithm that not only produces optimal makespan solutions, but
also is flexible enough to model real-world problems that adhere to the size limitations
defined by the abstract models in Chapter 5. While much research has been done on
multi-agent scheduling and many algorithms have been proposed, almost all of them
make many simplifying assumptions about the problem model or settle for sub-optimal
solutions by using heuristics. These heuristic techniques are useful for simplified
theoretical problem domains or for getting solutions quickly for large-scale scheduling
problems where it is acceptable for the solution to be within a certain bound of the
optimal value. In many cases though an optimal makespan may be sought for a smaller-
56
scale scheduling problem in business, military, or manufacturing domains, and it is these
cases that this thesis seeks to address.
4.1.1 Flexibility
Flexibility is a key issue in developing an algorithm to solve multi-agent scheduling
problems that occur in different domains, because each distinct problem may take a
different set of parameters into account. Many existing algorithms are designed to solve
specific classes of scheduling problems. For example, in some problems communication
delay may be so small compared to the execution delay that it can be considered
negligible, or jobs may not be precedence-constrained relative to each other. However,
the MILP formulation presented in this chapter seeks to solve not only R|PREC,rj|Cmax|JP
problems, the most complex class of problems defined by Lawler, Lenstra, Kan, and
McDowell [2,6], but also any simpler class of problem as well, and the problem may
include any of the following parameters:
• Communication delay can exist between agents, and may be a function of both the
data being communicated and the available bandwidth between a pair of agents.
• Execution delay for each job may be a function of both the specific job and the
agent that it is assigned to.
• Jobs may be precedence-constrained, and the precedence tree can take the form of
any directed, acyclic graph (DAG).
• Time in this model is continuous, not discrete-valued. Therefore execution and
communication delays may be any positive real-valued number and there is no
limit to the time horizon that the schedule must be performed within.
• Jobs may have release times as additional constraints. Release times specify the
earliest time that a job’ s execution can begin.
The tradeoff of having this flexibility and solution optimality is that there is a limit to the
problem size that the algorithm can handle, since scheduling is an NP –hard problem.
Therefore, the MILP formulation is most useful for solving schedules where there is
57
either a small number of agents, a small number of jobs, or where a large problem can be
decomposed recursively into a hierarchy of smaller scheduling problems, as described in
Section 3.6. Chapter 5 explores the limits to the size of scheduling problems that can be
feasibly solved with this algorithm.
4.1.2 Assumptions
While the MILP formulation is designed to be as flexible as possible to take many
different parameters into account, the nature of MILP techniques makes it necessary to
make a few simplifying assumptions in our problem model.
Assumption 1: Job pre-emption is not allowed
Once an agent begins to execute a job, it must continue to completion without
interruption. Also, an agent may process only one job at a time.
Assumption 2: All relevant parameters to the problem are known in advance
To compute the schedule to optimality, information about all available agents
including services offered, execution delays, and communication delays are
needed. If information about agents is not known a priori, those resources will
not be taken into account when solving the problem. Methods for obtaining this
information about the agent network are reviewed in Appendix C.
Assumption 3: The agent network is static
Because the entire schedule is computed prior to execution, any dynamic changes
in the agent organization or job precedence graph during execution of the
schedule may result in a sub-optimal or infeasible solution.
Assumption 4: Each individual job can be completed with a single agent
If a job requires a combination of services from multiple agents to be completed,
that job must be further decomposed into smaller tasks specific to each service
before the optimal schedule is computed, or that group of agents must be modeled
as a single agent.
58
While these assumptions place some limits on the types of problems that can be solved
using our technique, the algorithm is still flexible enough to be used for modeling
scheduling problems in many different scenarios.
4.2 Implementation
MILP techniques involve defining a set of constraints, decision variables, and an
objective function. The multi-agent scheduling problem can be modeled using the
following variables and constraints.
4.2.1 Problem Variables
Input Parameters Sets
J Set of all jobs j ∈ J that must be completed to achieve an objective. Each
job j is indexed numerically but may have a unique name that describes
the service needed to complete it.
A Set of all available agents a ∈ A. Each agent a is indexed numerically but
also may have a unique name associated with it.
Indicator Variables
P Immediate job precedence graph.
∈∈
=otherwise ,0
job precedesdirectly job if ,1, JkJjP kj
Q Full job precedence graph.
∈∈
=otherwise ,0
in job before anytime comes job if ,1, PJkJjQ kj
B Agent ability matrix
∈∈
=otherwise ,0
job complete toneeded services theprovides agent if ,1, JjAaB aj
59
Data Values
D Execution Delay matrix
D j,a = the execution delay for agent a∈ A to process job j ∈ J. ℜ∈ajD ,
C Communication Delay matrix
C j,a,b = the communication delay from sending the results of job j ∈ J
from agent a ∈ A to agent b ∈ A. To be more specific, we define C j,a,b as
a function of the number of bits that are produced by job j, bitsj,
transmitted from agent a to agent b with bandwidth, bandwidtha,b, in
bits/second. In this case, Cj,a,b = bitsj / bandwidtha,b. ℜ∈bajC ,,
R Job Release Times
R j = release time for job j ∈ J. ℜ∈jR
Decision Variables
Schedule Generation
X j,a Allocation of jobs to agents
∈∈
=otherwise ,0
job complete toassigned is agent if ,1,
JjAaX aj
S j Scheduled start time of job j ∈ J. ℜ∈jS
Support Variables
j,k Support variable used to determine whether jobs j ∈ J and k ∈ J overlap.
∈∈
=otherwise ,0
completed is job before started is job if ,1,
JjJkkjθ
M An upper bound on the makespan set by default to a very large number.
ℜ∈M
Objective
Cmax The makespan of the schedule. ℜ∈maxC
60
4.2.2 Constraints
The following linear-integer constraints describe the conditions needed to compute an
optimal multi-agent schedule.
Objective function:
Minimize: Cmax
Subject to:
{ }
∈
+≥
1,0
*
,
,,
max
aj
ajaj
j
X
XDSC 0, , >∈∈∀ ajBAaJj (4-1)
The objective function and agent/job allocation decision variables are defined in (4-1).
To minimize the number of irrelevant binary variables, Xj,a is only defined as binary
when B j,a > 0, i.e. when agent a offers the services necessary to complete job j.
0, =ajX 0, , =∈∈∀ ajBAaJj (4-2)
Constraint (4-2) states that an agent cannot be assigned a job unless it provides the
services necessary to complete that job. This constraint narrows down the total number
of binary decision variables that need to be computed so that an optimal solution is
reached faster.
1, =∑∈Aa
ajX Jj ∈∀ (4-3)
Constraint (4-3) specifies that each job must be assigned once to exactly one agent.
−+++≥
≥
)1(*)( ,,,,,
bkajbajaj
jk
jk
XXCDSS
SS
0,0,0,
),(,),(,,, >>>≠
∈∈∀bkajkj BBPkj
AbaJkj
(4-4)
61
The constraints in (4-4) state that a job cannot start until its predecessors are completed
and data has been communicated to it if the preceding jobs were executed on a different
agent.
j
j RS ≥ 0>∈∀ jRJj (4-5)
Constraint (4-5) specifies that a job cannot start until after its release time.
{ }
∈
−<−−
−≥−−
∑∑
∈
∈
1,0
)1(**
**
,
,,,
,,,
kj
kjjAa
ajaj
k
kjjAa
ajaj
k
MSXDS
MSXDS
θ
θ
θ
0,),( , =≠∈∀ kjQkjJkj (4-6)
3,,,, ≤+++ jkkjakaj XX θθ AaJkj ∈∈∀ ,),( (4-7)
Collectively, (4-6) and (4-7) specify that an agent may process at most one job at a time.
Constraint (4-7) states that if two jobs j and k are assigned to the same agent, their
execution times may not overlap. Mathematically, jobs j and k overlap if the following
two conditions are met:
1) jAa
ajaj
k SXDS +< ∑∈
,, * (4-8)
Job k starts sometime before job j finishes execution, and
2) kAa
akak
j SXDS +< ∑∈
,, * (4-9)
Job j starts sometime before job k finishes execution.
To describe these two conditions, the binary support variable j,k is defined to equal 1
when (4-8) is true, and 0 otherwise. Because (4-9) is equivalent to (4-8) with j and k
reversed, we can use the same variable k,j to describe (4-9). Therefore, jobs j and k
62
overlap when both j,k and k,j are true, or when� j,k + k,j > 1. Unfortunately, defining j,k
as a binary variable in terms of (4-8) and (4-9) is more difficult, because in an MILP
problem we cannot use decision variables such as Sj as part of the condition for a
constraint to occur. Instead, we must use the linear constraints in (4-6) to define j,k. If
M is set at a large enough value, j,k is forced to always assume the correct binary value
with relation to (4-8) and (4-9).
Together, these constraints define the multi-agent scheduling problem using a small
number of binary decision variables, which is an important factor in solving an MILP
problem efficiently. XpressMP™ optimization software is used to model these problems
and compute optimal schedules based on the above constraints (see Appendix B).
4.3 Multiple Objective Optimization
As shown in Chapter 3, an agent formulating a problem may often have a different
objective than minimizing the makespan of the schedule, or may have other objectives in
addition to it. These goals, like minimizing the makespan, can also be modeled using
mathematical objective functions. After determining the objective functions for each,
Goal Programming and Multiple Objective Linear Programming (MOLP) techniques
may be used to find the decision variable values whose solution satisfies the most
objectives.
4.3.1 Alternate Objective Functions
1) Minimize the average weighted completion time of all jobs
A common alternative to finding the optimal makespan of a schedule is to minimize the
average (weighted) completion time of all of the jobs. For example, in certain instances
it may be better to complete 4 out of 5 jobs very early even if the 5th job takes a long time
than to complete all of the jobs in less time, but have them all completed near the end of
the schedule. In this case we can either try to minimize the average completion time of
all jobs (weighing each job equally), or assign a fractional weight to each job signifying
63
how important that job is compared to other jobs. The latter case requires an additional
parameter:
W j = the fractional weight given to a particular job, where ℜ∈jW
and∑∈
=Jj
jW 1
The new objective function is
Minimize: ∑∑∈ ∈
+Jj Aa
jaj
ajj WXDS **)( ,
, (4-10)
2) Minimize the total number of agents used to solve the problem
In an agent environment it is often useful to allocate jobs to as few agents as possible that
can complete an objective in a reasonable amount of time. This is helpful when agents
are a scarce resource or are needed to be available to complete additional outside tasks,
and it makes the system more fault-tolerant if the structure of the agent network changes
dynamically. To model this objective an additional binary decision variable is defined:
∈
=otherwise ,0
complete tojobs moreor one assigned is agent if ,1 Aaaµ
a can be defined using integer-linear constraints in a similar fashion to j,k, based on the
binary decision values of Xj,a for each agent:
≤−
−−≥−
∑
∑
∈
∈
aJj
aj
aJj
aj
MX
MX
µ
µ
*5.0
)1(*5.0
,
,
Aa ∈∀ (4-11)
The objective function then becomes
Minimize: ∑∈Aa
aµ (4-12)
64
3) Minimize the total amount of bandwidth used between agents
Bandwidth is often a scarce resource in an agent network, so it may be beneficial to
minimize the total number of bits communicated between agents when computing an
optimal schedule. To calculate the number of communications that take place, a
composite binary decision variable Yj,a,k,b is defined:
otherwise ,0
1 and 1 if ,1 ,,,,,
==
= bkajbkaj
XXY
Using the following constraint:
bkajbkaj XXY ,,,,, *= (4-13)
The objective function then becomes:
Minimize: ∑ ∑≠
∈>
∈ba
AbaP
Jkj
jbkaj
kj
bitsY
|),(
0|),(
,,,
,
* (4-14)
4) Maximize the aggregate value associated with assigning jobs to particular agents
In many cases there may be a value associated with assigning tasks to particular agents.
For example, in a business model where each agent produces a different amount of
revenue by completing various tasks assigned to it, a manager may wish to maximize the
total revenue produced as an objective in computing the optimal schedule. In this case
the model requires an additional set of parameters:
value j,a = the value associated with assigning job j ∈ J to agent a ∈ A.
ℜ∈ajvalue ,
The objective function for maximizing the aggregate value is then defined as:
Maximize: ∑∑∈ ∈Jj Aa
ajaj Xvalue ,
, * (4-15)
65
5) Minimize the aggregate cost associated with assigning jobs to particular agents
Minimizing the costs associated with assigning tasks to agents is another common
objective that an agent may have when computing an optimal schedule. In a military
scenario, mobile autonomous agents use fuel to move around and may have risk levels
associated with performing different operations, and it is often beneficial to minimize
these values. Cost is defined by the following parameter:
cost j,a = the cost associated with assigning job j ∈ J to agent a ∈ A.
cost j,a ℜ∈
The objective function for minimizing cost is analogous to that for maximizing aggregate
value (4-15):
Minimize: ∑∑∈ ∈Jj Aa
ajaj Xstco ,
, * (4-16)
4.3.2 Multiple Objective Linear Programming
To optimize multiple objectives simultaneously using the same integer-linear program,
each objective is treated as a goal to be achieved. In many cases two objectives may
conflict with one another, thus making it impossible to obtain optimal values for both at
once. For example, in some problems minimizing the schedule’ s makespan may involve
using a greater number of agents and more bandwidth than desired, which would conflict
with any objective that tries to minimize costs associated with these parameters. In these
cases there is often no single optimal solution, and a value metric must be used as a
means of weighing possible solutions against each other. The problem may then be
solved iteratively until a satisfactory solution is produced [19].
To illustrate MOLP techniques, we use the multi-agent arithmetic example from Chapter
3, shown again in Figure 4-1. This time, however, we assume that there is a cost
associated with assigning a task to each agent (i.e. each agent charges a fee for every
operation that it performs). For simplicity we assume that the cost is specific to each
agent and is independent of the job chosen. Therefore, in addition to minimizing the
66
makespan of the schedule, Agent A has the additional objective of minimizing the total
cost across all agents. Suppose that cost is defined as:
COST = {$0, $4, $5, $3, $1, $2, $2} for each agent {A, B, C, D, E, F, G}
Figure 4-1: The agent organization and problem formulation used for the arithmetic example in Chapter 3
4.3.2.1 Determining Target Objective Values
The first step in solving an MOLP problem is to change each objective function into a
goal with a target value. The objective then becomes to find the optimal agent/job
allocation that achieves the closest approximation of these target values. It seems
reasonable that the optimal solution values for each of the individual objective functions
would provide good target values for the corresponding goals. Solving the MILP
problem twice, once for each individual objective function, produces the following
optimal target values:
Objective Function makespan cost
1) Minimize: Cmax 19s (target) $20
2) Minimize: ∑∑∈ ∈Jj Aa
ajaj Xstco ,
, * 26s $11 (target)
The objective now is to find the allocation that produces a makespan as close to 19s as
possible, and a total cost as close to $11 as possible.
Agent AServices/Delay:Initiate (I) / 1sTerminate (T) / 1s
Agent AServices/Delay:
Initiate (I) / 1sTerminate (T) / 1s
(5+8)*3 + (6-4)*(8+2)
Agent CServices/Delay:Multiply (*) / 2s
Agent CServices/Delay:Multiply (*) / 2s
Agent GServices/Delay:Add (+) / 1sMultiply (*) / 1s
Agent GServices/Delay:Add (+) / 1sMultiply (*) / 1s
Agent EServices/Delay:Subtract (-) / 1s
Agent EServices/Delay:Subtract (-) / 1s
Agent BServices/Delay:Add (+) / 3s
Agent BServices/Delay:Add (+) / 3s
Agent DServices/Delay:
Add (+) / 2s
Agent DServices/Delay:Add (+) / 2s
Agent FServices/Delay:Multiply (*) / 1s
Agent FServices/Delay:
Multiply (*) / 1s
4s
2s
1s
2s
2s
5s
2s
3s
Agent AServices/Delay:Initiate (I) / 1sTerminate (T) / 1s
Agent AServices/Delay:
Initiate (I) / 1sTerminate (T) / 1s
(5+8)*3 + (6-4)*(8+2)
Agent CServices/Delay:Multiply (*) / 2s
Agent CServices/Delay:Multiply (*) / 2s
Agent GServices/Delay:Add (+) / 1sMultiply (*) / 1s
Agent GServices/Delay:Add (+) / 1sMultiply (*) / 1s
Agent EServices/Delay:Subtract (-) / 1s
Agent EServices/Delay:Subtract (-) / 1s
Agent BServices/Delay:Add (+) / 3s
Agent BServices/Delay:Add (+) / 3s
Agent DServices/Delay:
Add (+) / 2s
Agent DServices/Delay:Add (+) / 2s
Agent FServices/Delay:Multiply (*) / 1s
Agent FServices/Delay:
Multiply (*) / 1s
4s
2s
1s
2s
2s
5s
2s
3s
Terminate7
Initiate0
+1 +3
+6
*4 *5
Problem Decomposition 1:(5+8)*3 + (6-4)*(8+2)
-2
Terminate7
Initiate0
+1 +3
+6
*4 *5
Problem Decomposition 1:(5+8)*3 + (6-4)*(8+2)
-2
67
4.3.2.2 Determining the Goal Programming Objective
The next step is to design a metric that can be used to weigh the two goals against each
other so that they are both optimized as much as possible. Because each of the target
goal values is measured in different units (dollars vs. time in this case), it is difficult to
compare tradeoffs between them directly. The fairest solution is to instead calculate the
percentage deviations of each goal’ s actual value from its target value and have the new
objective function be to minimize the weighted sum of these deviations:
Minimize: 11
11)*(*
1919
*,
,
2max
1
−+
− ∑∑∈ ∈Jj Aa
ajaj Xstco
wC
w (4-17)
By varying the values of w1 and w2, we can place different emphasis on each goal and
achieve different optimal allocations based on the relative importance of these goals to
one another. For example, by setting w1 and w2 each to 0.5, the schedule shown in Figure
4-2 is produced, which achieves the optimal target cost value ($11), and minimizes the
makespan to 23s, which is within 21% of its target value.
Time
Figure 4-2: An MOLP schedule that tries to minimize both the schedule’ s makespan and the execution cost on each agent, where w1 = .5 and w2 = .5. The makespan is 23s and the total cost is $11.
G
F
E
D
C
B
A
68
Varying w1 and w2 places different emphasis on each goal relative to the other one, which
results in different sets of optimal solutions, as shown in Figure 4-3. In this figure,
varying w1 and w2 between 0 and 1 produces three distinct solutions. The agent
formulating the problem must decide which of these solutions is best for each particular
scenario, and that can vary on a case-by-case basis.
Figure 4-3: A chart showing all of the different optimal solutions for minimizing the weighted sum of
percentage deviations, using different values of w1 and w2.
Each of the three possible solutions shown in Figure 4-3 represents a different corner
point of the MILP problem’ s feasible region. Minimizing the weighted sum of
percentage deviations from target values will only produce corner point solutions; in
order to explore other non-corner point solutions along the edge of the feasible region,
the MINIMAX objective function must be used instead. MINIMAX tries to minimize the
maximum percentage deviation of each individual goal from its target value instead of
minimizing a weighted sum of these values. A detailed description MINIMAX
optimization can be found in Spreadsheet Modeling and Decision Analysis, chapter 7
[19].
0.000.100.200.300.400.500.600.700.800.90
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
w1 value (emphasis on makespan)
perc
enta
ge d
evia
tion
from
ta
rget makespan
cost
23s/$11
20s/$16
19s/$20
(% deviationsfrom target)
0.000.100.200.300.400.500.600.700.800.90
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
w1 value (emphasis on makespan)
perc
enta
ge d
evia
tion
from
ta
rget makespan
cost
23s/$11
20s/$16
19s/$20
(% deviationsfrom target)
69
4.4 Refining the MILP Formulation
The amount of time needed to solve any particular MILP problem can vary significantly
depending on how well a problem is formulated. Unnecessary or redundant binary
decision variables or loose constraints may result in additional Branch and Bound
iterations and much longer computation times. Refining an MILP formulation can be
very difficult, and often involves a lot of trial and error due to the fact that there is no
guaranteed formula for making every model run faster. As shown in Chapter 2, the best
indicator of the quality of an MILP formulation is how well its constraints approximate
the convex hull of the feasible region. Unfortunately, this is very difficult to predict in
advance for all but the most trivial problems, as some problems involving only a hundred
decision variables may be very challenging, while others with thousands of variables are
sometimes solved with fewer iterations. While there are a few rules of thumb that will
generally lead to better problem formulations, the only way to find the best formulation is
to have insight into the structure of the system being modeled, and to run numerous
computational tests.
4.4.1 Eliminating Binary Decision Variables
The difference between Linear Programming and Mixed Integer-Linear Programming is
that for MILP problems the Branch and Bound technique must be used to recursively
refine solutions to optimality. This is a much more time-consuming process than just
using the Simplex method to solve an LP problem; in fact a new LP formulation must be
solved at each node in the Branch and Bound tree. When integer decision variables are
restricted to binary values, as they are in the MILP formulation presented in this chapter,
one of these variables is set to either 0 or 1 at each node of the Branch and Bound tree as
shown in Figure 4-4. Thus, in a worst-case scenario, a problem that has k binary decision
variables could have up to 2k Branch and Bound iterations. Fortunately this is almost
never the case because the Branch and Bound is a greedy algorithm that narrows down
possible solutions as it runs, but this shows that Branch and Bound problems can take
exponentially longer to solve as they use more binary variables.
70
Figure 4-4: A Branch and Bound tree with binary decision variables.
For this reason, reducing the number of binary variables used in an MILP problem is an
essential step in shortening computation time. The MILP formulation presented in this
chapter contains two binary decision variable types, Xj,a and j,k. A scheduling problem
that has m agents and n jobs contains (n*m) Xj,a variables and (n2) j,k variables.
Therefore, this MILP formulation would have n(n + m) total binary variables, and its
Branch and Bound tree would have at most 2n(n + m) nodes corresponding to these
variables.
4.4.1.1 Discrete Time vs. Continuous Time
The original MILP formulation developed for this thesis used a discrete-time model to
compute optimal multi-agent schedules. Instead of using the binary variable Xj,a and the
continuous variable Sj to describe the schedule, an additional parameter T and a binary
decision variable X j,a,t were used to represent job execution at discrete time values, where
∈∈∈
=otherwise ,0
at time job complete toassigned is agent if ,1,,
TtJjAaX taj
Using this discrete-time model made it much more straightforward to model the linear
constraints. For example, instead of having to define the support variable j,k to model
X1,1 = 0 X1,1 = 1
Branch and Bound Tree
X1,2 = 0 X1,2 = 0 X1,2 = 1X1,2 = 1
...
X1,1 = 0 X1,1 = 1
Branch and Bound Tree
X1,2 = 0 X1,2 = 0 X1,2 = 1X1,2 = 1
...
71
the restriction that each agent can only perform one service at a time (4-6, 4-7), this
specification can be modeled by a single linear constraint using discrete-time values:
2,,,, ≤+ saktaj XX ajakaj DtstBBkj
AaTstJkj,,, ,0,0,|
,),(,),(
+≤≤>>≠
∈∈∈∀ (4-18)
While using discrete-time variables simplifies the constraints needed to model the
problem, it results in a formulation that takes more time to solve and makes the algorithm
less flexible. Using discrete-time, all possible time values must be explicitly defined, and
therefore an upper bound on the schedule’ s makespan must be determined before the
problem is formulated. While this can be done using common heuristic techniques, it
limits the overall flexibility of the algorithm by forcing the time and
execution/communication delays to adhere to pre-defined interval lengths. Also,
discrete-time variables add another layer of combinatorial complexity to the problem:
instead of n(n+m) binary variables there are now (m*n*t) variables, where t is number of
distinct time steps in the model. Because of this added complexity, the MILP problem
was reformulated to use continuous time instead.
4.4.2 Refining Constraints
In general, adding more constraints to an MILP formulation tightens the bound on the
feasible region, thus creating a better approximation of the convex hull. While this tends
to result in fewer Branch and Bound iterations, adding more constraints also increases the
computational complexity of the LP relaxations produced at each step in the Branch and
Bound process. Therefore, it is important to weigh the tradeoffs in computation time
whenever a new set of constraints is added. For example, constraint (4-2) and the
constraint jk SS ≥ in (4-4) are not necessary to solve the scheduling problem to
optimality, but they help to create a tighter bound on the set of feasible points so that a
solution is reached faster.
72
Another improvement made to the MILP formulation in this chapter was to eliminate
unnecessary constraints that do not tighten the bound on the feasible region. The input
parameters B, P, and Q defined in Section 4.2.1.1 define each agent’ s abilities and the
structure of the job precedence tree, and are used to selectively apply constraints to only
the appropriate jobs or agents. For example, P and B are used to specify that the
constraints in (4-4) only apply to jobs that have immediate precedence constraints relative
to each other, and to agents that are capable of completing these jobs. Similarly, the
constraints in (4-6) state that two jobs may not be executed simultaneously on the same
agent, and Q is used to specify that these constraints only apply to agents that have no
precedence relations at all relative to each other.
0, =ajX 0, , =∈∈∀ ajBAaJj (4-2)
−+++≥
≥
)1(*)( ,,,,,
bkajbajaj
jk
jk
XXCDSS
SS
0,0,0,
),(,),(,,, >>>≠
∈∈∀bkajkj BBPkj
AbaJkj
(4-4)
{ }
∈
−<−−
−≥−−
∑∑
∈
∈
1,0
)1(**
**
,
,,,
,,,
kj
kjjAa
ajaj
k
kjjAa
ajaj
k
MSXDS
MSXDS
θ
θ
θ
0,),( , =≠∈∀ kjQkjJkj (4-6)
Creating tighter bounds on the integer feasible region and eliminating unnecessary
constraints and binary variables significantly shortens the time needed to compute an
optimal schedule using MILP techniques. For instance, consider the time taken to solve
the arithmetic scheduling problem from Chapter 3 using a discrete-time model versus a
continuous-time model, shown below in Figure 4-5.
73
Figure 4-5: Comparison of the number of constraints, decision variables, Branch and Bound iterations, and time to solve a scheduling problem using two different MILP formulations.
This figure shows that reducing the number of binary variables and modeling the MILP
formulation in continuous time reduces the problem to approximately 10% of its original
complexity. This difference becomes even greater with larger problems, as shown in the
next chapter.
0
1000
2000
3000
4000
5000
6000
Constraints Decisionvariables
Branch andBound
iterations
Time tosolve (ms)
discrete-time
continuous-time
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75
Chapter 5 Simulation and Results
This chapter examines the performance of the MILP formulation presented in Chapter 4
when it is applied to different abstract scheduling applications. As mentioned in Chapter
2, the main drawback of using MILP techniques is that the problem is NP-hard, and takes
exponential time to solve. Therefore, while the algorithm in this thesis can model the
most complex classifications of scheduling problems, solutions are limited to instances
where there is a small number of agents or jobs. By modeling the MILP problem using
XpressMP™ optimization software, we can gather data about the complexity of different
problem instances and determine the limits to the problem size that the simulation can
feasibly handle, and the rate at which complexity grows as more variables are added.
The first half of this chapter studies the complexity of the MILP scheduling formulation
with regard to the number of agents and jobs in the problem model. By fixing some
parameters and varying others, abstract examples are created that can be used as
benchmarks to measure the algorithm’ s performance relative to different combinations of
agents and jobs. The second half of this chapter examines a scheduling application based
on the closed-loop command and control architecture presented in Chapter 3 [10], and
76
studies how changing the agent organization for a given job precedence graph affects the
optimal makespan of the schedule as well as the difficulty of computing the makespan.
5.1 Scheduling Problem Complexity
This section explores the complexity of the MILP scheduling problem, based on the
number of agents and jobs used in each instance. As mentioned in Chapter 4, using a
continuous-time model instead of a discrete-time model allowed us to reduce the number
of binary variables to n(n+m) for a problem with n jobs and m agents. Based on this, it
would seem reasonable to assume that the complexity of the MILP problem depends
more on the number of jobs than on the number of agents. The following benchmarks
were used to test this theory and to determine the rate at which the exponential-time
algorithm runs relative to each of these two parameters. Each experiment was performed
on an Intel Pentium II processor running at 450MHz with 512Mb of available RAM
using XpressMP™ optimization software (see Appendix B).
5.1.1 Agent Variation
This benchmark demonstrates how the time to solve the MILP problem increases as the
number of agents increases. The number of jobs is fixed at one, and the execution delay
and communication delays between all agents are set to constant values (the agents are
assumed to be in a fully-connected network). The time to solve the problem as a function
of the number of agents is shown in Figure 5-1(a).
77
Single Job, Multiple Agents
(a) (b) Figure 5-1: Performance test of the time needed to solve the MILP scheduling problem with a single job
and multiple agents (a), and the number of constraints and variables used to model the problem, after presolving (b).
This data shows that when the number of jobs is limited to one, the complexity of the
problem still increases exponentially, but at a very slow rate. Even with over 1000
agents, an optimal schedule can still be computed in a reasonable amount of time (62s).
This makes sense because the problem has essentially been reduced to a resource
allocation problem where a single resource (job) must be allocated to the agent that
computes it the fastest, as there are no precedence constraints or parallel job executions to
take into account. Also, Figure 5-1(b) shows that the number of constraints and decision
variables grows linearly as a function of the number of agents.
However, the problem becomes significantly harder to solve once a second job is added,
as shown in Figure 5-2(a). With two precedence-constrained jobs, the time to compute
the optimal schedule increases at a much faster rate even with a smaller number of
agents. This is because with m agents, instead of m possible agent/job allocations, there
are now m2 different schedules. Even with this added complexity though, an optimal
schedule can still be computed for up to 90 agents in less than 10 minutes. It should be
noted that while the number of decision variables still grows linearly, the number of
constraints increases exponentially after the second job is introduced, which adds delay in
computing the LP relaxation of the problem at each Branch and Bound step.
0
10
20
30
40
50
60
70
0 200 400 600 800 1000
Number of Agents
Tim
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Number of Agents
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78
Two Jobs, Multiple Agents
(a) (b)
Figure 5-2: Performance test of the time needed to solve the MILP scheduling problem with two precedence-constrained jobs and multiple agents (a), and the number of constraints and variables used to
model the problem, after presolving (b).
5.1.2 Job Variation
This benchmark examines the scheduling algorithm’ s performance for different numbers
of precedence-constrained jobs, while keeping the number of agents fixed at one. The
jobs are ordered in a single precedence chain such that (job 1 % job 2 % … % job n), and
the execution and communication delays are constant values. Figure 5-3(a) shows the
performance of the MILP algorithm as a function of the number of jobs to be scheduled.
Multiple Jobs, Single Agent
(a) (b) Figure 5-3: Performance test of the time needed to solve the MILP scheduling problem with multiple precedence-constrained jobs and a single agent (a), and the number of constraints and variables used to
model the problem, after presolving (b).
0
200
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0 20 40 60 80 100
Number of Agents
Tim
e to
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0 50 100 150
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14000
0 50 100 150
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79
As seen by comparing Figures 5-3 and 5-1, the problem complexity increases at a higher
rate with the number of jobs than it does with the number of agents. This is because
while there is only one agent to schedule all of the jobs on, many more MILP constraints
are needed to enforce the precedence relations between the jobs, as shown in Figure 5-
3(b). Also, as mentioned earlier in this chapter, the number of binary decision variables
depends more on the number of jobs than on the number of agents. Using the MILP
scheduling algorithm, it is still possible to compute the schedule in less than 15 minutes
for up to 150 jobs with only a single agent.
Multiple Jobs, Two Agents
(a) (b) Figure 5-4: Performance test of the time needed to solve the MILP scheduling problem with multiple
precedence-constrained jobs and two agents (a), and the number of constraints and variables used to model the problem, after presolving (b).
As with the previous example, the MILP algorithm takes much longer to compute an
optimal schedule when another parameter is added to the problem. Adding another agent
to the model introduces communication delay, which increases the complexity of the
problem significantly. Figure 5-4(a) shows that the problem takes much longer to solve
with two agents, with the time to solve increasing exponentially until the number of jobs
reaches 27. With the 28th job, however, the algorithm’ s run time jumps from 691s to
27139.7s, an increase by a factor of almost 40. This sudden jump in computation time
may be caused by multiple factors. First, the XpressMP™ optimization software used to
solve this model sometimes employs different Branch and Bound node selection
strategies for the same problem model, resulting in varying computation times depending
on the nodes is chooses (see Appendix B). Also, the MILP Search Chart produced by
0
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15000
20000
25000
30000
0 5 10 15 20 25 30
Number of Jobs
Tim
e to
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(s)
MILP Run Time
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80
XpressIVE™ in Figure 5-5 shows that the lower bound produced by each successive LP
relaxation is increasing at a logarithmically when there are 28 jobs, and it will take a long
time for the lower bound to meet the upper bound at that rate. By inspection, however, it
can be seen in this figure that the optimal solution, 28s, was found in the first minute of
the algorithm’ s run time, so there is no need to let the algorithm run to completion in this
case.
Figure 5-5: An MILP search chart produced using XpressIVE™ software running a scenario with 2 agents
and 28 precedence-constrained jobs.
5.1.3 Job Parallelism
In addition to the number of jobs and agents used in the MILP model, the amount of
parallelism in the job precedence graph can also play a role in the complexity of the
scheduling problem. Being able to solve jobs in parallel across multiple agents not only
shortens the makespan of a schedule, but reduces the computation time of the optimal
schedule as well. Mathematically, the parallelism P of a job precedence graph can be
defined as the total execution time for all jobs in the system divided by the execution time
of the graph’ s critical path, as defined by Liao [16]:
P = )(max jL
D
Jj
Jj
j
∈
∈∑
(5-1)
81
In this equation L(j) is the sum of job execution times on the longest path from job j to an
exit task, a job that has no successors in the job precedence graph. Mathematically L(j) is
defined by Liao as:
∑∈∀
=k
k j
jDjLππ
max)( (5-2)
where kπ is the kth path from job j to an exit task.
To determine how job parallelism affects the complexity of the MILP scheduling
problem, we use the example from the previous section, two agents and multiple jobs, but
this time the jobs are precedence-constrained such that they form two parallel chains,
instead of a single chain. This configuration doubles the parallelism of the schedule, and
the optimal makespan can be reduced by 50% by assigning each chain to be executed in
parallel on a different agent. The time taken to compute the schedule is reduced as well,
as seen by comparing Figure 5-6 to Figure 5-4(a).
Multiple Parallel Jobs, Two Agents
Figure 5-6: Performance test of the time needed to solve the MILP scheduling problem with multiple precedence-constrained jobs and two agents when the jobs form two parallel chains.
0
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0 5 10 15 20 25 30
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Tim
e to
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(s)
MILP Run Time
82
5.1.4 Conclusion
While the MILP scheduling algorithm does not perform well with large-scale problems
because of its exponential complexity, these results show that the algorithm can be
effectively used to compute minimal makespans for scheduling problems where the
number of agents and jobs is kept small (<20). Also, by increasing the parallelism of the
job precedence graph, the computational time as well as the makespan can be greatly
reduced, allowing more agents and jobs to be modeled.
5.2 Simulation
The scenario addressed in this section is based on the closed-loop command and control
architecture developed at Draper Laboratory that was introduced in Chapter 3, Figure 3-1.
As mentioned in that chapter, this architecture is used to model autonomous systems and
can be used recursively to solve hierarchical problem decompositions. While the
architecture is used in Chapter 3 to describe the schedule optimization and multi-agent
problem-solving processes themselves, it can also be used to model a wide variety of
other agent planning and control scenarios, such as the hierarchical control of military air
operations [10].
The four sub-functions, Plan, Execute, Monitor, and Diagnose, can themselves be seen as
individual jobs that are precedence-constrained to each other by the flow of data between
them, as shown in Figure 5-6. For example, in this figure an autonomous system
generates a plan, and divides it into two sub-objectives which must themselves be
planned and executed, and the results are monitored, diagnosed, and returned to the
higher-level process. For simplicity, the feedback mechanism shown at each planning
stage in Chapter 3, Figure 3-3 has been eliminated, and it is assumed that the lower-level
planning processes simply forward their results back to the higher-level process without
immediately re-planning, and that only a single iteration of the closed-loop architecture is
performed.
83
Figure 5-6: Hierarchical Command and Control Architecture.
While it is often assumed that each 4-stage planning process takes place on a single
autonomous agent platform, these individual stages can actually be abstracted away and
seen as separate entities that can be computed independently by different agents. By
assigning each job a unique ID, the hierarchical command and control architecture can be
seen as a graph of precedence-constrained jobs that can be completed in parallel by a set
of agents that have the ability to perform these four operations.
Suppose that a commander wishes to use this architecture to complete a particular
problem, and that it has to choose the set of agents that can complete these ordered tasks
in the shortest amount of time. The commander has three sets of agents to choose from,
each of which has different execution delays for completing each job, and different
communication delays between each pair of agents. The three possible agent
organizations are shown in Figures 5-7(a),(b), and (c).
Plan (7)
Execute (8)Monitor (9)
Diagnose (10) Plan (3)
Execute (4)Monitor (5)
Diagnose (6)
Plan (1)
Execute (2)Monitor (11)
Diagnose (12)
Node 2 Node 3
Node 1
Plan (7)
Execute (8)Monitor (9)
Diagnose (10) Plan (3)
Execute (4)Monitor (5)
Diagnose (6)
Plan (1)
Execute (2)Monitor (11)
Diagnose (12)
Plan (7)
Execute (8)Monitor (9)
Diagnose (10) Plan (3)
Execute (4)Monitor (5)
Diagnose (6)
Plan (1)
Execute (2)Monitor (11)
Diagnose (12)
Node 2 Node 3
Node 1
84
Figure 5-7(a): Organization A. Two agents with equal capabilities at performing each job.
Figure 5-7(b): Organization B. Four agents that each specialize in performing a different job type.
Figure 5-7(c): Organization C. Three agents that specialize in each of the three nodes from Figure 5-6.
A1Execution TimesAll jobs: 2s
A1Execution TimesAll jobs: 2s
A2Execution TimesAll jobs: 2s
A2Execution TimesAll jobs: 2s
1sA1Execution TimesAll jobs: 2s
A1Execution TimesAll jobs: 2s
A2Execution TimesAll jobs: 2s
A2Execution TimesAll jobs: 2s
1s
B4Execution TimesDiagnose: 1sOther jobs: 3s
B4Execution TimesDiagnose: 1sOther jobs: 3s
B1Execution TimesPlan: 1sOther jobs: 3s
B1Execution TimesPlan: 1sOther jobs: 3s
B2Execution TimesExecute: 1sOther jobs: 3s
B2Execution TimesExecute: 1sOther jobs: 3s
B3Execution TimesMonitor: 1sOther jobs: 3s
B3Execution TimesMonitor: 1sOther jobs: 3s
1s
1s
1s
1s
B4Execution TimesDiagnose: 1sOther jobs: 3s
B4Execution TimesDiagnose: 1sOther jobs: 3s
B1Execution TimesPlan: 1sOther jobs: 3s
B1Execution TimesPlan: 1sOther jobs: 3s
B2Execution TimesExecute: 1sOther jobs: 3s
B2Execution TimesExecute: 1sOther jobs: 3s
B3Execution TimesMonitor: 1sOther jobs: 3s
B3Execution TimesMonitor: 1sOther jobs: 3s
1s
1s
1s
1s
C2Execution TimesNode 2 jobs: 1sOther jobs: 3s
C2Execution TimesNode 2 jobs: 1sOther jobs: 3s
C3Execution TimesNode 3 jobs: 1sOther jobs: 3s
C3Execution TimesNode 3 jobs: 1sOther jobs: 3s
3s
C1Execution TimesNode 1 jobs: 1sOther jobs: 3s
C1Execution TimesNode 1 jobs: 1sOther jobs: 3s
3s
C2Execution TimesNode 2 jobs: 1sOther jobs: 3s
C2Execution TimesNode 2 jobs: 1sOther jobs: 3s
C3Execution TimesNode 3 jobs: 1sOther jobs: 3s
C3Execution TimesNode 3 jobs: 1sOther jobs: 3s
3s
C1Execution TimesNode 1 jobs: 1sOther jobs: 3s
C1Execution TimesNode 1 jobs: 1sOther jobs: 3s
3s
85
Organization A contains two agents with a communication delay of 1s between them.
These agents don’ t specialize in any particular job, but can execute any job in 2s. Using
the MILP scheduling algorithm and XpressIVE™ to compute the optimal schedule for
this agent organization, the Gantt chart in Figure 5-8 is produced. The schedule is
computed in 2.5s, and the optimal makespan is 17s.
Figure 5-8: The optimal schedule computed for Organization A, with two agents performing 12 jobs.
Organization B, on the other hand, contains 4 agents, each of which is specialized to
perform a specific job type. Agent B1 takes 1s to perform the Plan operation, but 3s to
perform any other operation. Similarly, B2 specializes in Execution, B3 specializes in
Monitoring, and B4 specializes in Diagnosis. The Gantt chart produced by the MILP
scheduling algorithm in Figure 5-9 shows that by using this organization, the total
makespan can be reduced to 16s. Because there are more agents though, the MILP
problem is more complex and now takes 409s to compute the optimal schedule.
Agent
Time
86
Figure 5-9: The optimal schedule computed for Organization B, with four agents performing 12 jobs.
Organization C is the traditional agent organization that would be used in the command
and control hierarchy, where each agent corresponds to one of the 4-stage problem nodes.
These agents can perform any job in their assigned node in 1s, but take 3s to perform any
other job. However, this agent organization has longer communication delays between
agents, with each communication taking 3s. The optimal schedule using this agent
organization has an optimal makespan of 14s though, as shown in Figure 5-10, making it
the best choice out of the three.
Agent
Time
87
Figure 5-10: The optimal schedule computed for Organization C, with three agents performing 12 jobs.
Agent
Time
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89
Chapter 6 Conclusion
The goal of this thesis has been to explore multi-agent scheduling complexity and to
develop a technique for solving the most difficult classes of these problems. A high-
level, closed-loop framework is used to model the various stages of multi-agent problem-
solving, including problem formulation and decomposition, job/service mapping,
schedule computation, and task allocation. A new Mixed-Integer Linear Programming
(MILP) formulation is introduced which is capable of computing optimal makespans to
schedules in the presence of communication delay, job execution delay, release times,
and other parameters, and the algorithm’ s performance is studied under different
conditions. Extensions to the problem are also explored such as using MILP techniques
recursively in an agent command hierarchy to solve large scheduling problems, and using
Multiple Objective Linear Programming (MOLP) techniques to optimize additional
objective functions such as minimizing cost or bandwidth usage.
90
6.1 Original Contributions
This thesis presents a new, flexible MILP formulation that can be used to solve
scheduling problems with a variety of parameters in the context of a multi-agent
problem-solving framework. MILP techniques were chosen to model scheduling
problems because of the intuitive nature of modeling these problems as a set of
constraints with an objective function, and because these techniques produce optimal
solutions. A significant amount of work was done developing and optimizing this MILP
formulation by reducing the number of binary variables and redundant constraints used in
the model.
While the scheduling algorithm presented in this thesis is not suited to solve problems
with large numbers of agents and jobs, it can be used as an effective alternative to
existing heuristics if exact solutions are desired for complex classes of scheduling
problems that have small numbers of agents or jobs. The advantages and disadvantages
to using this MILP approach are listed below:
Advantages:
• The MILP approach presented in this thesis offers more flexibility than most
existing scheduling algorithms and heuristics. In addition to being able to solve
the most complex class of scheduling problems, R|PREC,rj|Cmax|JP, it can also
solve any simpler class of problems well. By taking into account communication
delay based on bandwidth between agents and the amount of data transmitted by
each job as well as execution delays on each agent, release times, and being able
to model any form of directed, acyclic job precedence graph, the algorithm can be
used in a wide variety of problem domains.
• The algorithm achieves completely optimal makespans for multi-agent scheduling
problems, unlike many heuristics that produce an approximate solution within a
certain bound of the true optimal value.
91
• In certain instances where there is an agent command hierarchy and a job
decomposition tree that has smaller granularity jobs at each successive level,
large-scale scheduling problems can be broken down into sets of smaller ones,
and the MILP scheduling algorithm can be applied recursively to produce optimal
solutions in a shorter amount of time.
• Using an MILP formulation to model the scheduling problem makes it possible to
easily change objective functions or to use Multiple Objective Linear
Programming (MOLP) techniques to achieve multiple objectives at once.
Minimizing the average weighted completion time of jobs, the number of agents
or amount of bandwidth used, or the aggregate cost of agent/job allocations may
be achieved in addition to minimizing the schedule’ s makespan.
• The MILP formulation presented in this thesis uses continuous-time instead of
discrete-time, thus reducing the number of binary variables needed to model the
problem and allowing execution and communication delays to assume any real-
number value, as well as creating an unbounded time horizon for the schedule.
Disadvantages:
• The main drawback to using an MILP algorithm is that because it runs in
exponential time it is only feasible to use on smaller-scale problems, where there
are a limited number of agents and jobs.
• Because the algorithm is static rather than dynamic, all information about the
agent network must be known a priori to solving the problem. Also, it is unable
to adapt the schedule to dynamic changes in the agent network or job precedence
graph during execution of the schedule itself; if a change is made the entire
schedule must be re-computed based on the new input data.
6.2 Future Work
Extensions to the work done in this thesis may include exploring a dynamic programming
approach to the same problem that is capable of taking into account uncertainty levels or
92
a limited view of the agent environment, and can adapt to unexpected changes in the
environment or problem domain during execution of the actual schedule. Other future
areas of research related to this thesis may also include dynamically re-organizing teams
of agents that respond to changing objectives and environments.
Problem formulation and decomposition techniques are also introduced in this thesis, but
no formula is developed to compute optimal job decompositions. Optimization
techniques to determine the best way to formulate and decompose a problem for a given
agent organization is another promising research avenue that may be pursued based on
the work done in this thesis, although in many cases the optimal decomposition seems to
vary depending on the nature of each individual problem.
Possible future extensions to the scheduling problem model itself and the MILP
formulation include allowing agents the ability to perform multiple jobs simultaneously,
and taking into account an agent’ s location and movement, allowing the scheduling
technique to be applied to vehicle scheduling and routing problems.
Additionally, it may be helpful to perform additional studies on the convergence rate of
the MILP technique relative to the number of agents and jobs in the problem, so that
approximate computation times may be predicted in advance and the Branch and Bound
algorithm may be stopped earlier with a near-optimal solution for large-scale problems.
It is important to note that while much research has been done on the complexity of
multi-agent scheduling problems and heuristics for solving them, considerable work
remains to develop an adaptive, flexible algorithm that is capable of taking into account
the numerous intricacies and details of today’ s real-world scheduling problems and
finding their optimal solutions.
93
Appendix A: MILP Formulation Minimize: Cmax Subject to:
{ }
∈
+≥
1,0
*
,
,,
max
aj
ajaj
j
X
XDSC 0, , >∈∈∀ ajBAaJj
0, =ajX 0, , =∈∈∀ ajBAaJj
1, =∑
∈AaajX Jj ∈∀
{ }
∈
−<−−
−≥−−
∑∑
∈
∈
1,0
)1(**
**
,
,,,
,,,
kj
kjjAa
ajaj
k
kjjAa
ajaj
k
MSXDS
MSXDS
θ
θ
θ
0,),( , =≠∈∀ kjQkjJkj
3,,,, ≤+++ jkkjakaj XX θθ AaJkj ∈∈∀ ,),(
−+++≥
≥
)1(*)( ,,,,,
bkajbajaj
jk
jk
XXCDSS
SS
0,0,0,
),(,),(,,, >>>≠
∈∈∀bkajkj BBPkj
AbaJkj
j
j RS ≥ 0>∈∀ jRJj
94
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95
Appendix B: XpressMP™ Optimization Software
The Mixed Integer-Linear Programming formulations presented in this thesis were
modeled and solved using Dash Optimization’ s XpressMP™ software. The software by
default uses Dual Simplex and Branch and Bound techniques to solve MILP problems,
and comes with its own proprietary modeling language, MOSEL. Using MOSEL it is
possible to model constraints, decision variables, and logical conditions without having to
explicitly construct a matrix of constraint coefficients.
B.1 Input File Format
Before the XpressMP™ Optimizer can be used to solve an MILP scheduling problem,
input parameters that define the agent organization and job precedence tree must be read
from a text file, such as the one shown in Figure B-1. This file contains the values for the
variables B j,a, D j,a, P j,k, Q j,k, and C j,a,b (or bandwidth a,b and bits j if C j,a,b is defined by
these values) and are modeled as matrices. If any of these parameters does not apply to a
particular problem, these matrices can be changed to reflect that. For example, if the
execution time for a job depended only on the agent performing it and not the job itself,
each row of D j,a would be identical. Similarly, if communication delay depended only
on the number of bits being transmitted, the bandwidth a,b matrix may contain all entries
of 1.
B.2 MOSEL Programming Language
MOSEL is a language designed specifically to solve Mathematical Programming
problems. Using MOSEL, MILP problems can be represented using linear constraints
and objective functions, just as they are in Chapter 4. The program then takes these
constraints, and forms a problem matrix, where each row represents a constraint, each
column represents a decision variable, and the matrix entries are the coefficients to these
variables. If none of the constraints or objective function uses inequalities such as ��RU����the entire problem can be solved using basic matrix solution techniques, treating the
96
Figure B-1: An example input file for a scheduling problem with 3 agents and 4 jobs
! total number of jobs agents and times NJ: 4 NT: 12 NA: 3 ! Durations of jobs on each agent ! D(i,j) is the execution time of job i on agent j D: [1 3.2 2.8 4 2 3 2 3.5 0.5 1.6 2 1.1] ! agents capable of performing each job ! B(i,j) = 1 means Job i can be completed by Agent j B: [1 1 0 1 1 1 0 1 1 1 1 0] ! Bandwidth between pairs of agents BAND: [1000 30 20 30 1000 50 20 50 1000] ! Number of bits transmitted as a result of each job BITS: [10 32 20 0] ! precedence constraints for jobs ! P(i j) means job i comes before job j P: [0 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0] ! full prec. tree: Q(i j) means job i comes at some point before job j Q: [0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 0]
97
problem as Ax = b, where A is the constraint coefficient matrix, x is a vector of decision
variables, and b is the objective function vector. However, the MILP formulation
presented in this thesis, along with most other existing LP problems, contains inequality
constraints, thus an algorithm such as the Simplex method must be used to obtain the
optimal solution of each LP relaxation.
An example of the MOSEL code used to solve the MILP formulation in Chapter 4 is
shown in Figure B-2. MOSEL allows the user to define sets of constraints based on
certain logical conditions, but it also contains basic programming constructs such as for
and while loops, and if/then statements. While most of the constraints listed in Chapter 4
can be translated directly to MOSEL code, there are a few that must be represented
differently. For example, the constraint
bkajbkaj XXY ,,,,, *= (4-13)
cannot simply be represented by the MOSEL code
forall(j in J, k in J, a in A, b in a) do Y(j,a,k,b) = X(j,a)*X(k,b) end-do
Multiplying Xj,a * Xk,b to produce Yj,a,k,b turns the problem into a quadratic programming
problem that would require more complicated methods to solve, so the following
auxiliary constraints are used instead to convey the same meaning.
forall(j in J, k in J, a in A, b in a) do Y(j,a,k,b) >= X(j,a) + X(k,b) - 1 Y(j,a,k,b) <= X(j,a) Y(j,a,k,b) <= X(k,b) Y(j,a,k,b) >= 0 end-do Because having these extra constraints adds to the overhead in solving the problem, it is
advantageous to avoid using extra binary decision variables such as Yj,a,k,b if possible.
98
Figure B-2: MOSEL code modeling the MILP constraints for the multi-agent scheduling problem.
...
... ! define objective function z forall(j in J, a in A | B(j,a)>0) do Cmax >= S(j) + D(j,a)*X(j,a) X(j,a) is_binary end-do ! a job can only be assigned to an agent capable of doing it forall(j in J, a in A | B(j,a) = 0) do X(j,a) = 0 end-do ! each job is done once on one agent forall(j in J) do SUM(a in A) X(j,a) = 1 end-do ! an agent may execute at most one job at a time forall(j in J, k in J | (j<>k AND Q(j,k)=0)) do S(k) - (S(j) + SUM(a in A) (X(j,a)*D(j,a))) >= -M*O(j,k) S(k) - (S(j) + SUM(a in A) (X(j,a)*D(j,a))) <= M*(1 - O(j,k)) - R O(j,k) is_binary end-do forall(j in J, k in J, a in A) do O(j,k) + O(k,j) + X(j,a) + X(k,a) <= 3 end-do ! a job cannot start until its predecessors are completed forall(j in J, k in J, a in A, b in A | (j<>k AND P(j,k)>0 AND B(j,a)>0 AND B(k,b)>0)) do (S(k) - S(j))/(D(j,a) + C(j,a,b)) >= (X(j,a) + X(k,b) - 1) (S(k) - S(j))/(D(j,a) + C(j,a,b)) >= 0 end-do ! objective function minimize(Cmax) ... ...
99
B.3 Xpress™ Optimizer
Once a Mathematical Programming formulation is compiled, the XpressMP™ Optimizer
is used to compute the optimal solution to the problem. After the problem matrix has
been constructed, a presolve technique is performed which is used to eliminate redundant
constraints and variables. In the case of MILP problems, bound tightening and
coefficient tightening are applied to tighten the LP relaxation as part of the presolve. For
example, if the matrix has the binary variable x and one of the constraints is x <= 0.2,
presolve will automatically fix x at 0 because it can never take the value 1. Presolve
produces a new, smaller problem matrix which is then used to solve the LP relaxation of
the MILP problem.
Using these LP relaxations and the Branch and Bound technique, the XpressMP™
Optimizer solves for the optimal integer solution by tightening its upper bound (the best
integer solution found) and the lower bound (found using the LP relaxation) until these
bounds meet, as in Figure B-3. It should be noted that while solutions are usually
consistent, the Optimizer sometimes chooses to pursue different paths in the Branch and
Bound tree for different instances of the same problem model, resulting in different
computation times and numbers of nodes traversed when the model is run multiple times.
Figure B-3: Output from Xpress-IVE showing the upper and lower bounds of the optimal integer solution
to an example MILP scheduling problem and the time taken to find this solution.
100
B.4 Graphing Functionality
XpressIVE™, the programming interface used to model problems in MOSEL and display
solutions, features graphing functionality which was very useful for displaying optimally
computed schedules in the form of a Gantt chart. In these charts, the x-axis is used to
represent time in the schedule, and the y-axis represents each individual agent. Each
horizontal numbered arrow in the graph represents a different job and the time needed to
execute that job, and the arrows that connect these jobs to each other represent the
communication of job data between agents, and the delay caused by this. Figure B-4
shows an example of a schedule computed in a scenario where there are 4 agents and 16
jobs that are executed in parallel.
Figure B-4: A Gantt chart produced using the graphing function in Xpress-IVE™
In addition to this graphical solution, Xpress-IVE™ also displays the total number of
constraints and decision variables before and after presolve, the number of Simplex
iterations needed to solve the LP relaxation, the number of nodes traversed during the
Branch and Bound process, and the time needed to solve the problem to optimality. This
data is useful for determining the limits to the problem size that could be solved in a
reasonable amount of time, and for comparing different agent and job configurations to
each other.
Time
Agent
101
Appendix C: Agent/Service Discovery
In order to optimally solve a multi-agent scheduling problem using the algorithm
presented in this thesis, it is necessary to first have full information about the agent
environment and the resources that are available to solve the problem. Therefore, a
planning agent must discover the structure of the physical agent network, including all
services offered by various agents, their execution times, and communication delays
before determining how to allocate jobs to agents. Considerable research has been done
on inter-agent communication and agent discovery techniques that can help an agent to
formulate a scheduling problem [6].
C.1 Agent Directories
Because the infrastructure of an agent environment is often dynamic, with agents entering
and leaving at random intervals, it is helpful to have a central system set up that keeps
track of which agents are available at any given time. This data repository may be used
as a reference for agents entering the network that wish to collect data about other
available agents and services, or to provide updates for agents already in the network.
One such method developed by Decker, Sycara, and Williamson [6] is to use middle
agents, whose sole purpose is to collect data about available resources in the network and
advertise this data to any agent that requests it. This data consists of a directory of white
pages and yellow pages. White pages are simply a listing of all agents currently in the
network, and the address used to identify and communicate with that agent, such as an IP
address. White pages may contain additional information such as available bandwidth
between each pair of connected agents. Yellow pages are a directory of services
provided by all agents in the network, and may contain a reference to the corresponding
white pages entry for each agent, and possibly a listing of execution delays for each
service. White pages and yellow pages may be kept up-to-date by having the middle
agent send multicast requests and periodic pings to agents in the network. By using both
white page and yellow page services, agents can keep track of the services offered by
other agents and communication and execution delays for each of these services.
102
C.2 Finding the Shortest Path Between Agents
An agent may be able to determine the available bandwidth and communication delay
between agents that are directly connected to it in the network simply by querying these
agents directly. However, in order to compute an optimal schedule it is necessary to
know shortest or least-costly path between all pairs of agents in the network, including
those not directly connected to each other. In other words, given an adjacency matrix of
communication delays between connected agents, an agent needs to compute a shortest-
path matrix between all pairs of agents in the network. Figure C-1 shows the adjacency
and shortest-path matrices for the network described in Chapter 3, Figure 3-5 below:
035304
0252022
203302
4220
G
F
E
D
C
B
A
GFEDCBA
⇒
0375787307676477024545620232774203386533027442320
G
F
E
D
C
B
A
GFEDCBA
(a) (b) Figure C-1: Adjacency matrix (a) and shortest-path matrix (b) showing the communication delay between
agents for the network in Chapter 3, Figure 3-5.
Fortunately, efficient algorithms are available for computing the shortest-path matrix
from an adjacency matrix. The most widely known of these is Dijkstra’ s Algorithm [18].
Given a graph G with a set of nodes N and a set of edges E, each edge with a cost we,
Dijkstra’ s Algorithm computes the shortest path (the path with lowest total cost) from a
starting node s ∈ N to every other node in N.
Dijkstra’ s algorithm works by keeping track of an array A of best-known estimates for
each node’ s distance from the starting node, and a priority queue of these nodes ordered
by this estimated distance. The algorithm starts by looking at the starting node and all of
103
the nodes directly adjacent to it. The adjacent node closest to the starting node is
removed from the front of the priority queue, and the nodes adjacent to that node are
examined and A is updated to reflect any shorter paths that are found. This process is
repeated, each time removing the node with the closest estimated distance to the starting
node, until there are no nodes left. The distance calculated in A for each node that is
removed from the priority queue is the shortest path between it and the starting node. A
pseudo-code representation of Dijkstra’ s Algorithm is shown below, derived from Morris
[18]:
initialize_single_source(Graph g, Node s) { for each node n in Nodes(g) g.distance[n] = infinity g.pi[n] = nil g.distance[s] = 0; } relax(Node u, Node v, cost[][]) { if distance[v] > distance[u] + cost[u][v] then distance[v] = distance[u] + cost[u][v] pi[v] = u } dijkstra(Graph g, Node s) { initialize_single_source(g,s) S = {0} Q = Vertices(g) // priority queue while Q is not empty u = ExtractCheapest(Q); add node u to S; for each vertex v adjacent to u relax(u,v,cost[u][v]) }
The most efficient way to implement Dijkstra's Algorithm is to use adjacency lists to
store the graph and to use a heap as a priority queue. If the graph has m edges and n
nodes, then the algorithm runs O((m+n)log n) time. Because for the agent networks
presented in this thesis we want to find the distance from every node to every other node
though, this algorithm must be run n times and therefore the total running time is
O((m+n)*nlog n). Once the complete shortest-path matrix has been computed, these
communication delays may be used to compute the optimal job allocation among agents
in the network.
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105
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